Research Article Parareal Algorithms Implemented with IMEX …downloads.hindawi.com/journals/mpe/2015/395340.pdf · IMEX ( im, ex,*) to identify scheme ,where im is the number of

Research ArticleParareal Algorithms Implemented withIMEX Runge-Kutta Methods

Zhiyong Wang1 and Shulin Wu2

1School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 610731 China2School of Science Sichuan University of Science and Engineering Zigong Sichuan 643000 China

Correspondence should be addressed to Zhiyong Wang uestczhywanggmailcom

Received 17 June 2014 Revised 31 August 2014 Accepted 1 September 2014

Academic Editor Zhike Peng

Copyright copy 2015 Z Wang and S WuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Parareal algorithm is a very powerful parallel computation method and has received much interest frommany researchers over thepast few years The aim of this paper is to investigate the performance of parareal algorithm implemented with IMEX Runge-Kutta(RK) methods A stability criterion of the parareal algorithm coupled with IMEX RKmethods is established and the advantage (inthe sense of stability) of implementing with this kind of RK methods is numerically investigated Finally numerical examples aregiven to illustrate the efficiency and performance of different parareal methods

1 Introduction

The parareal algorithm was presented by Lions et al in [1]as a numerical method to solve time dependent problems inparallel The peculiarity of this algorithm is that it approx-imates successfully the solution later in time before havingfully accurate approximations from earlier time points Thisalgorithm is becoming more and more popular in scientificand engineering computation andmany excellent results havebeen obtained

As mentioned above the parareal algorithm was firstintroduced in [1] and an improved version which aims tosolve nondifferentiable PDEswas discussed byBal andMadayin [2] Some further modifications and improvements can befound in [3] inwhich the authors use the ldquofilteringrdquo techniqueto overcome the so-called resonance or beating phenomenon(see also [4 5]) that triggers numerical instability whenthe algorithm is applied to linear structure dynamics Thestability and convergence are the main subjects of theoreticalanalysis of the algorithmwhich have been investigated widelyby many researchers see for example [6ndash8] Nowadays thisalgorithm as well as its variants [3 9ndash12] has been usedin many fields by many researchers such as morphologicaltransformation simulations [13] structural (fluid) dynamics

simulations [4 5 14] optimal control [15 16] Hamiltoniansimulations [17 18] turbulent plasma simulations [19 20]and solution of Volterra integral equations [21] (There is anincreasing interest in parareal and it is very possible thatsome important references are not mentioned here)

For ODEs

1199101015840

(119905) = 119891 (119905 119910 (119905)) 119905 isin [0 119879]

119910 (0) = 1199100

(1)

to formulate the parareal algorithm the time domain [0 119879]

of interest is first partitioned into several time-slices whoseboundary points are treated as coarse time-grids And thenthe algorithm consists of three basic steps First using somenumerical propagator sayG

Δ119879 the solution is approximated

on each coarse time-grid to provide a seedmdashthat is an initialconditionmdashto the time-slice Second another propagatordenoted by F

Δ119905 is applied independently and therefore

concurrently in each time-slice to advance the solution fromthe starting point of this time-slice to its end point Finallyan iterative process is invoked to improve the accuracy ofthe seeds and eliminate the jumps of the solution on thecoarse time-grids In most cases the numerical propagatorsGΔ119879

and FΔ119905

are defined by traditional RK method with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 395340 12 pageshttpdxdoiorg1011552015395340

2 Mathematical Problems in Engineering

equal length of the time-slices such as Radau methodsLobatto methods and Gauss methods and there are lots ofinvestigations about theoretical properties and performanceof practical application of the algorithm in such case See forexample [4 5 7 9 14] and references therein

There is also some research about special choice of theunderlying numerical methods that are used to define G

Δ119879

and FΔ119905 For example Guibert and Tromeur-Dervout [22]

considered the adaptive time-slices (ie the length of time-slices is not equal) to define G

Δ119879and F

Δ119905and the derived

parareal algorithm can be applied to strong stiff ODEs anddifferential algebraic equations (DAEs) Another exampleis the one investigated by Minion [11] where the authorsdefined the numerical propagators G

Δ119879and F

Δ119905by spec-

tral deferred correction based on the Gaussian quadraturespectral integration It was shown that the iterative processbased on spectral deferred correction rather than traditionalmethods within a parareal framework results in a significantdecrease in the overall computational cost of the algorithmWu et al [9] suggested that the propagator F

Δ119905was defined

by local Richardson extrapolation and it turns out thatthe Parareal-Richardson method has advantages of higheraccuracy and better stability

In some cases the function 119891 in (1) can be written intotwo parts

119891 = 119891119904+ 119891119899119904 (2)

where 119891119899119904

is the nonstiff or mildly stiff part of 119891 and 119891119904is

the stiff part A common example for such case arises fromsemidiscretization of reaction diffusion equations119906

119905= 119889119906119909119909+

119903(119905 119909 119906) In this case the function 119891 in (1) can be written as119891(119905 119910(119905)) = 119860119910(119905) + 119891

119899119904(119905 119910(119905)) with 119860 being a tridiagonal

matrix and stiff In the case that 119891 can be partitioned intostiff and nonstiff parts it is more advisable to use the implicit-explicit (IMEX) RKmethods (or additive RK methods calledsometimes) to solve ODEs (1) For this type of RK methodsthe stiff part is assigned with the implicit component ofthe IMEX RK method to satisfy stability requirement andthe nonstiff or mildly stiff part is assigned with the explicitcomponent to reduce computational cost For more detailsabout the RK methods and the IMEX RK methods theinterested reader may refer to [23ndash33] and references therein

Therefore under condition (2) we think that it is valuableto adopt the IMEX RKmethods instead of the traditional RKmethods to define G

Δ119879andor F

Δ119905used in parareal frame-

work This is the main motivation of our work In certainaspects this paper should be viewed as an experimental oneWhile the stability of the derived algorithm we present iseasily proven by the results given by Gander and Vandewalle[7] stability region of the parareal algorithm defined by theIMEX RK methods is numerically shown

The structure of this paper is as follows In Section 2the parareal algorithm and the IMEX RK methods arerevisited Section 3 is concerned with the stability ofthe derived parareal algorithm Several combinations of

different IMEX RK methods and the correspondingadvantagesdisadvantages are numerically shown Finally inSection 4 we test the Gray-Scott model arising in chemicalreaction to illustrate our results

2 The Parareal Algorithm andthe IMEX RK Methods

In this section we revisit the parareal algorithm and theIMEX RK methods

21 The Parareal Algorithm To introduce the parareal algo-rithm let us first partition time interval [0 119879] into 119873

subintervals 119878119899= [119879119899 119879119899+1

] 119899 = 0 1 119873 minus 1 with equallength Δ119879 and 119879

119899= 119899Δ119879 We call 119879

119899rsquos the coarse time-grids

We then use some finer step size Δ119905 = Δ119879119872 (119872 gt 1 is aninteger) to partition each relative large interval 119878

119899into119872finer

intervals 119904119899119898

= [119879119899+119898119872

119879119899+(119898+1)119872

]with119879119899+119898119872

= 119879119899+119898Δ119905

and 119898 = 0 1 119872 minus 1 Now the parareal algorithm canbe described as follows We designate by symbol ⊖ the time-sequential steps performed on the coarse time-grids and bysymbol oplus the parallel steps performed on the decomposedfiner time-grids

The Parareal Algorithm Consider the following

⊖ Step 0 (initialization) Perform sequential computation1198840

119899+1= GΔ119879(119879119899 1198840

119899 Δ119879) with 119884

0

0= 1199100 119899 = 0 1 119873 minus 1

for 119896 = 0 1

oplus Step 1 Perform in subinterval 119878119899= [119879119899 119879119899+1

] the computa-tion

119899+(119898+1)119872= FΔ119905(119879119899+119898119872

119899+119898119872

Δ119905) with initial value119884119896

119899119898 = 0 1 119872 minus 1

⊖ Step 2 Perform sequential corrections 119884119896+1119899+1

= GΔ119879(119879119899

119884119896+1

119899 Δ119879) +

119899+1minus GΔ119879(119879119899 119884119896

119899 Δ119879) with 119884

119896+1

0= 1199100 119899 = 0

1 119873 minus 1

⊖ Step 3 If for 119899 = 0 1 119873 minus 1 119884119896+1119899

satisfy some termina-tion criteria stop the iteration else go to Step 1

Here the finer propagation operator FΔ119905

and the coarsepropagation operatorG

Δ119879relay on one-step numericalmeth-

ods and if the underlying methods are implicit in most cases(linear ODEs is an exception) nonlinear algebraic equationsneed to be solved

The above algorithm can be written compactly as

119884119896+1

119899+1= GΔ119879

(119879119899 119884119896+1

119899 Δ119879) +F

Δ119905(119879119899 119884119896

119899 Δ119905)

minusGΔ119879

(119879119899 119884119896

119899 Δ119879)

(3)

where 119896 is the iteration index Note that for 119896 rarr +infinmethod(3) will generate a series of values 119884

119899which satisfy 119884

119899+1=

FΔ119905(119879119899 119884119899 Δ119905) This implies that the converged solution 119884

119899

will achieve the accuracy of the propagatorFΔ119905

Mathematical Problems in Engineering 3

22 The IMEX RK Methods Let us consider a pair of twoRunge-Kutta methods defined by the arrays

0 0 0 0 sdot sdot sdot 0

1198882

11988621

11988622

0 sdot sdot sdot 0

1198883

11988631

11988632

11988633

0

d 0

119888119904

1198861199041

1198861199042

sdot sdot sdot 119886119904119904minus1

119886119904119904

1198871

1198872


119887119904


1198882

11988621

0 0 sdot sdot sdot 0

1198883

11988631

11988632

0 0

d 0

119888119904

1198861199041

1198861199042


0

1

2


119904

(4)

with the same abscissae

119888119894=

119894

sum

119895=1

119886119894119895=

119894minus1

sum

119895=1

119886119894119895 119894 = 2 119904 (5)

The top formula of (4) determines a diagonally implicit(semi-implicit) RK method and the bottom formula is anexplicit RK method In addition let ℎ gt 0 be a step sizeand define the step point 119905

119899= 119899ℎ for integer 119899 Consider the

following stiffnonstiff partitioned ODEs

1199101015840

(119905) = 119891119904(119905 119910 (119905)) + 119891

119899119904(119905 119910 (119905)) 119905 isin [0 119879]

119910 (0) = 1199100

(6)

and by applying the top part of (4) to stiff component 119891119904and

the bottom part to the nonstiff component we obtain thefollowing scheme

119870119894= 119910119899+ ℎ

119894

sum

119895=1

119886119894119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119894minus1

sum

119895=1

119886119894119895119891119899119904(119905119899+ 119888119895ℎ119870119895) 119894 = 1 2 119904

119910119899+1

= 119910119899+ ℎ

119904

sum

119895=1

119887119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119904

sum

119895=1

119895119891119899119904(119905119899+ 119888119895ℎ119870119895)

(7)

where sum119894119895=1

(sdot) = 0 with 119894 lt 119895The order conditions for IMEX RK methods are very

complicated and beyond the topic of this paper For thisaspect we refer the interested reader to for example [26 30]At the moment we introduce some frequently used IMEXmethods with order 119901 = 1 2 3 We will use the tripletIMEX(119904im 119904ex 119901) to identify scheme (4) where 119904im is thenumber of stages of the implicit part 119904ex is the number ofstages of the explicit part and 119901 is the order of the IMEXscheme

First Order IMEX RK Methods The most popular first orderIMEX RK method is the IMEX Euler method

0 0 0

1 0 1

0 1

0 0 0

1 1 0

1 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX Euler

(8)

SecondOrder IMEXRKMethodsThe second order IMEXRKmethods considered here are the IMEX trapezoidal scheme[34]

0 0 0

11

2

1

2

1

2

1

2

0 0 0

1 1 0

1

2

1

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX trapezoidal

(9)

The IMEX trapezoidal scheme is a combination of thetrapezoidal rule andHeunrsquos second ordermethod (the explicittrapezoidal rule)

Third Order IMEX RK Methods For third order IMEX RKmethod we consider the one constructed by Ascher et al [23](this scheme is denoted by IMEX(4 4 3))

0 0 0 0 0

119886 0 119886 0 0

07179332607 0 02820667392 119886 0

1 0 12084966490 minus0644363171 119886

0 12084966490 minus0644363171 119886

0 0 0 0

119886 0 0 0

03212788860 03966543747 0 0

minus01058858296 119887 119887 0

0 12084966490 minus0644363171 119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX(443)

(10)

where 119886 = 04358665215 and 119887 = 0552929147 Both theimplicit and explicit schemes of IMEX(4 4 3) are third orderRK methods and the implicit scheme is 119871-stable

To finish this section we introduce the concept of stabilityof the IMEX RK methods which will be used in the stabilityanalysis of the parareal algorithm


The asymptotic behavior of the IMEX RK methodsapplied to (6) is analyzed on the basis of the scalar modelequation

1199101015840

(119905) = 120582119910 + 120583119910 120582 120583 isin C (11)

which was proposed by Frank et al [35] and Verwer andSommeijer [36] and adopted by Koto [28 37] For modelequation (11) from (7) we arrive at

119870 = E119910119899+ ℎ120582119860119870 + ℎ120583119860119870

119910119899+1

= 119910119899+ ℎ120582b119870 + ℎ120583b119870

(12)

where we have used vector andmatrix notations to denote thevalues shown in the Butcher tableau (4) that is119860 = (119886

119894119895)119860 =

(119860119894119895) b = (119887

1 1198872 119887

119904) and b = (

1 2

119904) Moreover

E = (1 1 1)119879

Let 120572 = ℎ120582 120573 = ℎ120583 We then have

119910119899+1

= 119877 (120572 120573) 119910119899 (13)

where 119877(120572 120573) is called stability function of the IMEX RKmethod which is defined by

119877 (120572 120573) = 1 + (120572b + 120573b) (119868 minus 120572119860 minus 120573119860)minus1

E (14)

where 119868 is an identity matrix of size 119904

3 Stability of the Parareal Algorithm

In our analysis of the stability of the parareal algorithmimplemented with IMEX RK methods an important role isa strictly lower triangular Toeplitz matrixM = M(120590) of size119873 Its elements are defined by

M119894119895=

0 if 119894 le 119895

120590119894minus119895minus1

if 119894 gt 119895(15)

The next necessity for our analysis is the estimation ofM119896

infin which is proved by Gander and Vandewalle [7]

Lemma 1 (see [7]) Let 119896 gt 0 be an integer Then the infinitenorm ofM119896 can be estimated by

10038171003817100381710038171003817M119896

(120590)10038171003817100381710038171003817infin

le

min(1 minus |120590|119873minus1

1 minus |120590|)

119896

(119873 minus 1

119896) 119894119891 |120590| lt 1

|120590|119873minus119896minus1

(119873 minus 1

119896) 119894119891 |120590| ge 1

(16)

Assume that the stability functions of the underlyingnumerical methods used to define the two propagators G

Δ119879

and FΔ119905

are 119877F(120572 120573) and 119877G(120572 120573) respectively Then itfollows by applying the parareal algorithm to the modelequation (11) that

GΔ119879

(119879119899 119884119896

119899 Δ119879) = 119877G (120572 120573) 119884

119896

119899

FΔ119905(119879119899 119884119896

119899 Δ119879) = 119877

119872

F (120572

119872120573

119872)119884119896

119899 119896 = 1 2

(17)

where we recall119872 = Δ119879Δ119905

Theorem 2 Let Δ119879 be given and 119879119899

= 119899Δ119879 for 119899 =

0 1 Let the underlying numerical method used to definethe propagator G

Δ119879running on the coarse time-grids be in its

region of absolute stability that is |119877G| lt 1 Then one has thefollowing estimates

sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 119884119896

119899

10038161003816100381610038161003816

le (

10038161003816100381610038161003816119877119872

F (120572119872 120573119872) minus 119877G (120572 120573)10038161003816100381610038161003816

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 1198840

119899

10038161003816100381610038161003816

119896 ge 1

(18)

Proof We denote by 119890119896119899the error at iteration 119896 of the parareal

algorithm at coarse time point 119879119899 that is 119890119896

119899= 119910(119905

119899) minus 119884119896

119899

Then with an induction argument on 119899 it follows by applyingthe iterative process (3) to the model equation (11) that thiserror satisfies

119890119896+1

119899+1= 119877G (120572 120573) 119890

119896+1

119899+ [119877F (

120572

119872120573

119872) minus 119877G (120572 120573)] 119890

119896

119899

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]

119899

sum

119895=1

119877119899minus119895

G(120572 120573) 119890

119896

119895

(19)

where we have used the fact that 1198901198960= 0 for any 119896 ge 0 Set

119864119896= (119890119896

1 119890119896

2 119890

119896

119873)119879 Then relation (19) can be written in

matrix form as

119864119896+1

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]M (119877G (120572 120573)) 119864

119896

(20)

where thematrixM is defined by (15) with 120590 = 119877G(120572 120573)Thisimplies

119864119896= [119877F (

120572

119872120573

119872) minus 119877G(120572 120573)]

119896

M119896(119877G (120572 120573)) 119864

0 (21)


Therefore from Lemma 1 and the assumption |119877G| lt 1 wehave

1003817100381710038171003817100381711986411989610038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times10038171003817100381710038171003817M119896(119877G (120572 120573))

10038171003817100381710038171003817

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times (1 minus

1003816100381610038161003816119877G (120572 120573)1003816100381610038161003816119873minus1

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

(22)

Hence the proof of this theorem is completed by letting119873 rarr

+infin in the second inequality of (22)

The factor 120588(120572 120573) = |119877119872

F (120572119872 120573119872) minus 119877G(120572 120573)|(1 minus

|119877G(120572 120573)|) is called by Gander and Vandewalle [7] thelinear convergence factor of the parareal algorithm performedon unbounded time intervals According to the conceptsintroduced by Bal [6] and Staff and Roslashnquist [8] it is alsocalled the stability function of the parareal algorithm In thispaper we use the latter name

Define

D = (120572 120573) | 120588 (120572 120573) lt 1 120572 120573 isin C (23)

Then if (120572 120573) isin D the parareal algorithm is convergent onunbounded time intervals In what follows of this paper thesetD is called stability region of the parareal algorithm

Remark 3 If b = b and 119860 = 119860 (ie the IMEX RK methodsreduce to the traditional RK methods) it follows from (14)that 119877(120572 120573) = R(119911) = 1 + 119911b(119868 minus 119911119860)

minus1E with 119911 = 120572 + 120573

Therefore if both the propagators GΔ119879

and FΔ119905

are definedby the traditional RK methods the stability function 120588(120572 120573)

can be rewritten as120588(119911) = |R119872F(119911119872)minusRG(119911)|(1minus|RG(119911)|)which is the one obtained by Gander and Vandewalle in [7]

From Remark 3 we see that the underlying numericalmethod used to define propagator G

Δ119879that is in its absolute

stability region is necessary for the stability of the pararealalgorithm In fact in the previous analysis and applicationof the parareal algorithm (ie both the propagators G

Δ119879

and FΔ119905

are defined by the traditional RK methods) it hasbeen proposed by Farhat et al [4 5 14] that the underlyingRK method for G

Δ119879should be implicit and the one for the

propagator FΔ119905

should be explicit The reason is obviousmdashthe implicit RK method is used to guarantee stability and theexplicit RK method is used to reduce computation cost asmuch as possible

The greatest advantage of using an IMEX RK methodis that the stability of the combined scheme approaches tothat of its implicit part while the computational cost can bearcomparison with simply using the explicit part Hence in theparareal framework it is naturally desired that the adoptionof the IMEX RK methods will provide a far more stableparareal algorithm with computational cost sharply reduced

To see whether our desirability can be carried through fortwo IMEX RK methods IMEXG and IMEXF we considerhere 119872 = 50 and the following four special combinationswhich leads to four parareal algorithms

PIM-EXGΔ119879 is defined by the implicit part of IMEXG

andFΔ119905is defined by the explicit part IMEXF

PIM-IMEX GΔ119879 is defined by the implicit componentof IMEXG andF

Δ119905is defined by IMEXF

PIMEX-EX GΔ119879 is defined by IMEXG and FΔ119905

isdefined by the explicit component of IMEXF

PIMEX-IMEX GΔ119879 is defined by IMEXG and FΔ119905

isdefined by IMEXF

For the convenience of plotting the stability region of theparareal algorithm we consider the case (120572 120573) = (119909 119894119910)119909 119910 isin R The stability functions of the above four pararealalgorithms are denoted by RIM-EX(119909 119910) RIM-IMEX(119909 119910)RIMEX-EX(119909 119910) and RIMEX-IMEX(119909 119910) respectively The sta-bility regions are denoted by DIM-EX DIM-IMEX DIMEX-EXand DIMEX-IMEX respectively We remark that the PIM-EXalgorithm is a conventional parareal algorithm and we willcompare its stability region with the ones of the other threealgorithms

We first illustrate the stability regions of the parareal algo-rithms when IMEXG is chosen as the IMEX Euler method(see (8)) For IMEXF = IMEX Euler and IMEXF = IMEXTrapezoidal and IMEXF = IMEX(4 4 3) the stability regionof the algorithmsPIM-EXPIM-IMEXPIMEX-EX andPIMEX-IMEXis plotted in Figures 1(a) 1(b) and 1(c) respectively ForIMEXG = IMEX Euler the stability functions of the pararealalgorithms shown in Figures 1(a) and 1(b) are

Figure 1(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus ((119872 + 119894119910) (119872 minus 119909))

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816

RIMEX-IMEX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus ((119872 + 119894119910) (119872 minus 119909))

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816


0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300

400119967IM-IMEX 119967IMEX-EX 119967IMEX-IMEX119967IM-EX

minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10

minus100 minus100 minus500 0

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)

Figure 1 (a) (IMEXG IMEXF)= (IMEXEuler IMEXEuler) (b) (IMEXG IMEXF)= (IMEXEuler IMEXTrapezoidal) (c) (IMEXG IMEXF)

= (IMEX Euler IMEX(4 4 3))

Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)

Figure 2 (a) (IMEXG IMEXF) = (IMEX Trapezoidal IMEX Euler) (b) (IMEXG IMEXF) = (IMEX Trapezoidal IMEX Trapezoidal)(IMEXG IMEXF) = (IMEX Trapezoidal IMEX(4 4 3))

where 119872 = 50 For the case IMEXF = IMEX(4 4 3) thestability functions of the four parareal algorithms are verycomplex and the presentations are omitted

From Figure 1 we see that the stability region of thePIMEX-IMEX algorithm seems only a little smaller than that ofthe PIM-IMEX algorithm and both are significantly larger thanthat of the PIM-EX algorithm We note that the computationcost of PIM-IMEX is obviously more expensive than that ofPIMEX-IMEX since in some cases solving nonlinear equationsis not involved in the PIMEX-IMEX algorithm Therefore weget the conclusion that for IMEXG = IMEX Euler it is betterto use the PIMEX-IMEX algorithm instead of PIM-IMEX (in thesense of computational cost) and PIM-EX (in the sense ofstability) Moreover it is clear that the computational costof the PIMEX-EX algorithm is the least one among the fouralgorithmswhile fromFigure 1we see that the stability regionof this algorithm is the smallest one and particularly it issmaller than that of the PIM-EX algorithm However from thepoint of computational cost of view this algorithm still standscomparison with PIM-EX

We next consider the case IMEXG = IMEX Trapezoidalscheme (see the left scheme of (9)) For IMEXF = IMEXEuler and IMEXF = IMEX Trapezoidal and IMEXF =

IMEX(4 4 3) the stability regions of the four algorithmsPIM-EX PIM-IMEX PIMEX-EX and PIMEX-IMEX are plotted inFigures 2(a) 2(b) and 2(c) respectively For IMEXG = IMEXTrapezoidal the stability functions of the parareal algorithmsshown in Figures 2(a) and 2(b) are

Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910minus (1 +

119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)

Figure 3 Behavior of the solutions of the Gray-Scott model (26)-(27)

RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910minus (

119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909minus (

119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)

where 119872 = 50 Again for the case IMEXF = IMEX(4 4 3)we omit the presentations of the four stability functionsbecause of high complexity


0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3

times10minus3 times10

minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7

Figure 4 Measured error diminishing of the solutions 119906 and V when the PIMEX-EX algorithm is used

For IMEXG = IMEX Trapezoidal from Figure 2 we seethat the results are different from the previous case IMEXG =IMEX Euler and are more interesting (a) the stability regionof each algorithm shrinks to a smaller region (b) for each lalgorithm the stability regions for different IMEXF are verysimilar (c) it seems that there is no difference betweenPIM-EX(PIMEX-EX) and PIM-IMEX (PIMEX-IMEX)

Based on the results shown in Figures 1 and 4 we get thefollowing two conclusions

(1) it is more advisable to use IMEX Euler method asthe underlying numerical method for the G

Δ119879

propagator instead of using the IMEX Trapezoidalmethod


5 10 15 20 25 30

10minus10

10minus5

100

105

PIMEX-IMEXPEX-IMEXPIMEX-EX

(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02

minus100 minus50minus40

minus30

minus20

minus10


minus40

minus30

minus20

minus10

119967EX-IMEX 119967IMEX-EX 119967IMEX-IMEX

(b)

Figure 5 Measured error diminishing (a) and stability region (b) of the algorithms PEX-IMEX PIMEX-EX and PIMEX-IMEX

(2) if IMEXG = IMEX Trapezoidal it seems that thePIM-EX algorithm is the best one since it outperformsPIMEX-EX and PIMEX-IMEX in the sense of stability andoutperforms PIM-IMEX in the sense of computationalcost

4 Numerical Results

In this section we show some numerical results to illustratethe good performance of the parareal algorithm implementedwith the IMEX RK method We test the well-known Gray-Scott model arising from chemical reaction (see [38 39])

119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)

where 119905 isin [0 20] 119909 isin [0 1] and 1205981= 10minus4 1205982= 10minus6 120578 =

0024 and 120579 = 006 The initial-boundary condition for (26)is chosen as

119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)

With these conditions we expect three pulses in the solutionsof (26) Figure 3 shows a typical behaviour of the solutions

The PDE system (26) is first discretized spatially as 119906119909119909

asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1

minus2V119895+V119895minus1

)Δ1199092 with

Δ119909 = 001 119895 = 1 2 100 and then a nonlinear ordinarydifferential system that consists of 200 ODEs is solved by theparareal algorithmWe use the IMEX Euler method to defineboth the G

Δ119879propagator and F

Δ119905propagator We consider

here 119872 = 50 Δ119879 = 110 and Δ119905 = Δ119879119872 In Figure 4 weshow the first 6 iterations of the parareal PIMEX-EX algorithmwhere we see that the error diminishing for the solutions 119906and V is very rapid

We also test the convergence speed of the pararealPIMEX-EX algorithm and the parareal algorithm which isdenoted by PEX-IMEX with propagators G

Δ119879and F

Δ119905being

defined by the explicit Euler and the IMEX Euler methodsrespectively The convergence curves corresponding to thesethree parareal algorithms are shown in Figure 5(a) We seethat the PEX-IMEX algorithm is not convergent and both theother two algorithms converge rapidlyThis obversion can beexplained by the stability region shown in Figure 5(b) wherewe see that the stability region of PEX-IMEX is significantlysmaller than that of the other two algorithms Moreoveras has been shown in Figure 1 we see that the stabilityregion of the PIMEX-EX algorithm is nearly contained inthe one of PIMEX-IMEX This means that for the Gray-Scottmodel (26)-(27) if the PIMEX-EX algorithm converges sodoes the PIMEX-IMEX algorithm Moreover it is interestingthat the PIMEX-EX algorithm converges a little sharper thanPIMEX-IMEX see Figure 5(a)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors are very grateful to the anonymous referees forthe careful reading of a preliminary version of the paperand their valuable suggestions and comments which reallyimprove the quality of this paper This work was sup-ported by Science amp Technology Bureau of Sichuan Province(2014JQ0035) Non-Destructive Inspection of Sichuan Insti-tutes of Higher Education (2013QZY01) the project spon-sored by OATF-UESTC and the NSF of China (1130105811301362 11371157 and 91130003)

References

[1] J Lions Y Maday and G Turinici ldquoA ldquopararealrdquo in timediscretization of PDErsquosrdquo Comptes Rendus de IrsquoAcademie desSciences Series I Mathematics vol 332 no 1 pp 661ndash668 2001

[2] G Bal and Y Maday ldquoA ldquopararealrdquo time discretization for non-linear PDErsquos with application to the pricing of anAmerican putrdquoin Recent Developments in Domain Decomposition Methods LF Pavarino and A Toselli Eds vol 23 of Lecture Notes inComputational Science and Engineering pp 189ndash202 2002

[3] M J Gander and M Petcu ldquoAnalysis of a Krylov subspaceenhanced parareal algorithmrdquo ESAIM Proceedings vol 29 no2 pp 556ndash578 2007

[4] J Cortial and C Farhat ldquoA time-parallel implicit method foraccelerating the solution of non-linear structural dynamicsproblemsrdquo International Journal for Numerical Methods inEngineering vol 77 no 4 pp 451ndash470 2009

[5] C Farhat J Cortial C Dastillung and H Bavestrello ldquoTime-parallel implicit integrators for the near-real-time prediction oflinear structural dynamic responsesrdquo International Journal forNumerical Methods in Engineering vol 67 no 5 pp 697ndash7242006

[6] G Bal ldquoOn the convergence and the stability of the pararealalgorithm to solve partial differential equationsrdquo in DomainDecomposition Methods in Science and Engineering vol 40 ofLecture Notes in Computational Science and Engineering pp425ndash432 Springer Berlin Germany 2005

[7] M J Gander and S Vandewalle ldquoAnalysis of the parareal time-parallel time-integration methodrdquo SIAM Journal on ScientificComputing vol 29 no 2 pp 556ndash578 2007

[8] G A Staff and E M Roslashnquist ldquoStability of the pararealalgorithmrdquo in Domain Decomposition Methods in Science andEngineering vol 40 of Lecture Notes in Computational Scienceand Engineering pp 449ndash456 Springer Berlin Germany 2005

[9] S Wu B Shi and C Huang ldquoParareal-Richardson algorithmfor solving nonlinear ODEs and PDEsrdquo Communications inComputational Physics vol 6 no 4 pp 883ndash902 2009

[10] M Emmett and M L Minion ldquoToward an efficient parallelin time method for partial differential equationsrdquo Communica-tions in Applied Mathematics and Computational Science vol 7no 1 pp 105ndash132 2012

[11] M L Minion ldquoA hybrid parareal spectral deferred correctionsmethodrdquoCommunications in AppliedMathematics and Compu-tational Science vol 5 no 2 pp 265ndash301 2010

[12] X Dai and Y Maday ldquoStable parareal in time method forfirst- and second-order hyperbolic systemsrdquo SIAM Journal onScientific Computing vol 35 no 1 pp A52ndashA78 2013

[13] L P He and M X He ldquoParareal in time simulation ofmorphological transformation in cubic alloys with spatially

dependent compositionrdquo Communications in ComputationalPhysics vol 11 no 5 pp 1697ndash1717 2012

[14] C Farhat and M Chandesris ldquoTime-decomposed paralleltime-integrators theory and feasibility studies for fluid struc-ture and fluid-structure applicationsrdquo International Journal forNumerical Methods in Engineering vol 58 no 9 pp 1397ndash14342003

[15] Y Maday J Salomon and G Turinici ldquoMonotonic pararealcontrol for quantum systemsrdquo SIAM Journal on NumericalAnalysis vol 45 no 6 pp 2468ndash2482 2007

[16] T P Mathew M Sarkis and C E Schaerer ldquoAnalysis ofblock parareal preconditioners for parabolic optimal controlproblemsrdquo SIAM Journal on Scientific Computing vol 32 no3 pp 1180ndash1200 2010

[17] X Dai C le Bris F Legoll and Y Maday ldquoSymmetric pararealalgorithms for Hamiltonian systemsrdquo ESAIM MathematicalModelling and Numerical Analysis vol 47 no 3 pp 717ndash7422013

[18] M J Gander and E Hairer ldquoAnalysis for parareal algorithmsapplied to Hamiltonian differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 259 no part A pp 2ndash13 2014

[19] J M Reynolds-Barredo D E Newman R Sanchez D Samad-dar L A Berry and W R Elwasif ldquoMechanisms for theconvergence of time-parallelized parareal turbulent plasmasimulationsrdquo Journal of Computational Physics vol 231 no 23pp 7851ndash7867 2012

[20] J M Reynolds-Barredo D E Newman and R Sanchez ldquoAnanalytic model for the convergence of turbulent simulationstime-parallelized via the parareal algorithmrdquo Journal of Com-putational Physics vol 255 pp 293ndash315 2013

[21] X Li T Tang and C Xu ldquoParallel in time algorithm withspectral-subdomain enhancement for Volterra integral equa-tionsrdquo SIAM Journal on Numerical Analysis vol 51 no 3 pp1735ndash1756 2013

[22] D Guibert and D Tromeur-Dervout ldquoParallel adaptive timedomain decomposition for stiff systems of ODEsDAEsrdquo Com-puters amp Structures vol 85 no 9 pp 553ndash562 2007

[23] U M Ascher S J Ruuth and R J Spiteri ldquoImplicit-explicitRunge-Kutta methods for time-dependent partial differentialequationsrdquoAppliedNumericalMathematics An IMACS Journalvol 25 no 2-3 pp 151ndash167 1997

[24] S Sekar ldquoAnalysis of linear and nonlinear stiff problemsusing the RK-Butcher algorithmrdquo Mathematical Problems inEngineering vol 2006 Article ID 39246 15 pages 2006

[25] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995

[26] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983

[27] M Mechee N Senu F Ismail B Nikouravan and Z Siri ldquoAthree-stage fifth-order Runge-Kutta method for directly solvingspecial third-order differential equationwith application to thinfilm flow problemrdquoMathematical Problems in Engineering vol2013 Article ID 795397 7 pages 2013

[28] T Koto ldquoIMEX Runge-Kutta schemes for reaction-diffusionequationsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 182ndash195 2008


[29] Q Ming Y Yang and Y Fang ldquoAn optimized Runge-Kuttamethod for the numerical solution of the radial Schrodingerequationrdquo Mathematical Problems in Engineering vol 2012Article ID 867948 12 pages 2012

[30] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006

[31] X Duan J Leng C Cattani and C Li ldquoA Shannon-Runge-Kutta-Gill method for convection-diffusion equationsrdquo Math-ematical Problems in Engineering vol 2013 Article ID 163734 5pages 2013

[32] L Pareschi and G Russo ldquoImplicit-Explicit Runge-Kuttaschemes and applications to hyperbolic systems with relax-ationrdquo Journal of Scientific Computing vol 25 no 1-2 pp 129ndash155 2005

[33] H Yuan C Song and P Wang ldquoNonlinear stability and con-vergence of two-step Runge-Kutta methods for neutral delaydifferential equationsrdquo Mathematical Problems in Engineeringvol 2013 Article ID 683137 14 pages 2013

[34] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations SpringerBerlin Germany 2003

[35] J Frank W Hundsdorfer and J G Verwer ldquoOn the stability ofimplicit-explicit linear multistep methodsrdquo Applied NumericalMathematics vol 25 no 2-3 pp 193ndash205 1997

[36] J G Verwer and B P Sommeijer ldquoAn implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equationsrdquoSIAM Journal on Scientific Computing vol 25 no 5 pp 1824ndash1835 2004

[37] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 no 2 pp 201ndash212 2008

[38] J E Pearson ldquoComplex patterns in a simple systemrdquo Sciencevol 261 no 5118 pp 189ndash192 1993

[39] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


equal length of the time-slices such as Radau methodsLobatto methods and Gauss methods and there are lots ofinvestigations about theoretical properties and performanceof practical application of the algorithm in such case See forexample [4 5 7 9 14] and references therein

There is also some research about special choice of theunderlying numerical methods that are used to define G

Δ119879

and FΔ119905 For example Guibert and Tromeur-Dervout [22]

considered the adaptive time-slices (ie the length of time-slices is not equal) to define G

Δ119879and F

Δ119905and the derived

parareal algorithm can be applied to strong stiff ODEs anddifferential algebraic equations (DAEs) Another exampleis the one investigated by Minion [11] where the authorsdefined the numerical propagators G

Δ119879and F

Δ119905by spec-

tral deferred correction based on the Gaussian quadraturespectral integration It was shown that the iterative processbased on spectral deferred correction rather than traditionalmethods within a parareal framework results in a significantdecrease in the overall computational cost of the algorithmWu et al [9] suggested that the propagator F

Δ119905was defined

by local Richardson extrapolation and it turns out thatthe Parareal-Richardson method has advantages of higheraccuracy and better stability

In some cases the function 119891 in (1) can be written intotwo parts

119891 = 119891119904+ 119891119899119904 (2)

where 119891119899119904

is the nonstiff or mildly stiff part of 119891 and 119891119904is

the stiff part A common example for such case arises fromsemidiscretization of reaction diffusion equations119906

119905= 119889119906119909119909+

119903(119905 119909 119906) In this case the function 119891 in (1) can be written as119891(119905 119910(119905)) = 119860119910(119905) + 119891

119899119904(119905 119910(119905)) with 119860 being a tridiagonal

matrix and stiff In the case that 119891 can be partitioned intostiff and nonstiff parts it is more advisable to use the implicit-explicit (IMEX) RKmethods (or additive RK methods calledsometimes) to solve ODEs (1) For this type of RK methodsthe stiff part is assigned with the implicit component ofthe IMEX RK method to satisfy stability requirement andthe nonstiff or mildly stiff part is assigned with the explicitcomponent to reduce computational cost For more detailsabout the RK methods and the IMEX RK methods theinterested reader may refer to [23ndash33] and references therein

Therefore under condition (2) we think that it is valuableto adopt the IMEX RKmethods instead of the traditional RKmethods to define G

Δ119879andor F

Δ119905used in parareal frame-

work This is the main motivation of our work In certainaspects this paper should be viewed as an experimental oneWhile the stability of the derived algorithm we present iseasily proven by the results given by Gander and Vandewalle[7] stability region of the parareal algorithm defined by theIMEX RK methods is numerically shown

The structure of this paper is as follows In Section 2the parareal algorithm and the IMEX RK methods arerevisited Section 3 is concerned with the stability ofthe derived parareal algorithm Several combinations of

different IMEX RK methods and the correspondingadvantagesdisadvantages are numerically shown Finally inSection 4 we test the Gray-Scott model arising in chemicalreaction to illustrate our results

2 The Parareal Algorithm andthe IMEX RK Methods

In this section we revisit the parareal algorithm and theIMEX RK methods

21 The Parareal Algorithm To introduce the parareal algo-rithm let us first partition time interval [0 119879] into 119873

subintervals 119878119899= [119879119899 119879119899+1

] 119899 = 0 1 119873 minus 1 with equallength Δ119879 and 119879

119899= 119899Δ119879 We call 119879

119899rsquos the coarse time-grids

We then use some finer step size Δ119905 = Δ119879119872 (119872 gt 1 is aninteger) to partition each relative large interval 119878

119899into119872finer

intervals 119904119899119898

= [119879119899+119898119872

119879119899+(119898+1)119872

]with119879119899+119898119872

= 119879119899+119898Δ119905

and 119898 = 0 1 119872 minus 1 Now the parareal algorithm canbe described as follows We designate by symbol ⊖ the time-sequential steps performed on the coarse time-grids and bysymbol oplus the parallel steps performed on the decomposedfiner time-grids

The Parareal Algorithm Consider the following

⊖ Step 0 (initialization) Perform sequential computation1198840

119899+1= GΔ119879(119879119899 1198840

119899 Δ119879) with 119884

0

0= 1199100 119899 = 0 1 119873 minus 1

for 119896 = 0 1

oplus Step 1 Perform in subinterval 119878119899= [119879119899 119879119899+1

] the computa-tion

119899+(119898+1)119872= FΔ119905(119879119899+119898119872

119899+119898119872

Δ119905) with initial value119884119896

119899119898 = 0 1 119872 minus 1

⊖ Step 2 Perform sequential corrections 119884119896+1119899+1

= GΔ119879(119879119899

119884119896+1

119899 Δ119879) +

119899+1minus GΔ119879(119879119899 119884119896

119899 Δ119879) with 119884

119896+1

0= 1199100 119899 = 0

1 119873 minus 1

⊖ Step 3 If for 119899 = 0 1 119873 minus 1 119884119896+1119899

satisfy some termina-tion criteria stop the iteration else go to Step 1

Here the finer propagation operator FΔ119905

and the coarsepropagation operatorG

Δ119879relay on one-step numericalmeth-

ods and if the underlying methods are implicit in most cases(linear ODEs is an exception) nonlinear algebraic equationsneed to be solved

The above algorithm can be written compactly as

119884119896+1

119899+1= GΔ119879

(119879119899 119884119896+1

119899 Δ119879) +F

Δ119905(119879119899 119884119896

119899 Δ119905)

minusGΔ119879

(119879119899 119884119896

119899 Δ119879)

(3)

where 119896 is the iteration index Note that for 119896 rarr +infinmethod(3) will generate a series of values 119884

119899which satisfy 119884

119899+1=

FΔ119905(119879119899 119884119899 Δ119905) This implies that the converged solution 119884

119899

will achieve the accuracy of the propagatorFΔ119905




1198882

11988621

11988622

0 sdot sdot sdot 0

1198883

11988631

11988632

11988633

0

d 0

119888119904

1198861199041

1198861199042


119886119904119904

1198871

1198872


119887119904


1198882

11988621


1198883

11988631

11988632

0 0

d 0

119888119904

1198861199041

1198861199042


0

1

2


119904

(4)


119888119894=

119894

sum

119895=1

119886119894119895=

119894minus1

sum

119895=1

119886119894119895 119894 = 2 119904 (5)




1199101015840

(119905) = 119891119904(119905 119910 (119905)) + 119891

119899119904(119905 119910 (119905)) 119905 isin [0 119879]

119910 (0) = 1199100

(6)



119870119894= 119910119899+ ℎ

119894

sum

119895=1

119886119894119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119894minus1

sum

119895=1

119886119894119895119891119899119904(119905119899+ 119888119895ℎ119870119895) 119894 = 1 2 119904

119910119899+1

= 119910119899+ ℎ

119904

sum

119895=1

119887119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119904

sum

119895=1

119895119891119899119904(119905119899+ 119888119895ℎ119870119895)

(7)

where sum119894119895=1




0 0 0

1 0 1

0 1

0 0 0

1 1 0

1 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX Euler

(8)


0 0 0

11

2

1

2

1

2

1

2

0 0 0

1 1 0

1

2

1

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX trapezoidal

(9)



0 0 0 0 0

119886 0 119886 0 0

07179332607 0 02820667392 119886 0

1 0 12084966490 minus0644363171 119886

0 12084966490 minus0644363171 119886

0 0 0 0

119886 0 0 0

03212788860 03966543747 0 0

minus01058858296 119887 119887 0

0 12084966490 minus0644363171 119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX(443)

(10)





1199101015840

(119905) = 120582119910 + 120583119910 120582 120583 isin C (11)


119870 = E119910119899+ ℎ120582119860119870 + ℎ120583119860119870

119910119899+1

= 119910119899+ ℎ120582b119870 + ℎ120583b119870

(12)


119894119895)119860 =

(119860119894119895) b = (119887

1 1198872 119887

119904) and b = (

1 2

119904) Moreover

E = (1 1 1)119879


119910119899+1

= 119877 (120572 120573) 119910119899 (13)



E (14)




M119894119895=

0 if 119894 le 119895

120590119894minus119895minus1

if 119894 gt 119895(15)




10038171003817100381710038171003817M119896

(120590)10038171003817100381710038171003817infin

le


1 minus |120590|)

119896

(119873 minus 1

119896) 119894119891 |120590| lt 1

|120590|119873minus119896minus1

(119873 minus 1

119896) 119894119891 |120590| ge 1

(16)


Δ119879

and FΔ119905


GΔ119879

(119879119899 119884119896

119899 Δ119879) = 119877G (120572 120573) 119884

119896

119899

FΔ119905(119879119899 119884119896

119899 Δ119879) = 119877

119872

F (120572

119872120573

119872)119884119896

119899 119896 = 1 2

(17)



= 119899Δ119879 for 119899 =




sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 119884119896

119899

10038161003816100381610038161003816

le (

10038161003816100381610038161003816119877119872

F (120572119872 120573119872) minus 119877G (120572 120573)10038161003816100381610038161003816

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 1198840

119899

10038161003816100381610038161003816

119896 ge 1

(18)



119899= 119910(119905

119899) minus 119884119896

119899


119890119896+1

119899+1= 119877G (120572 120573) 119890

119896+1

119899+ [119877F (

120572

119872120573

119872) minus 119877G (120572 120573)] 119890

119896

119899

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]

119899

sum

119895=1

119877119899minus119895

G(120572 120573) 119890

119896

119895

(19)


119864119896= (119890119896

1 119890119896

2 119890

119896


matrix form as

119864119896+1

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]M (119877G (120572 120573)) 119864

119896

(20)


119864119896= [119877F (

120572

119872120573

119872) minus 119877G(120572 120573)]

119896

M119896(119877G (120572 120573)) 119864

0 (21)



1003817100381710038171003817100381711986411989610038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times10038171003817100381710038171003817M119896(119877G (120572 120573))

10038171003817100381710038171003817

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times (1 minus

1003816100381610038161003816119877G (120572 120573)1003816100381610038161003816119873minus1

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

(22)



The factor 120588(120572 120573) = |119877119872

F (120572119872 120573119872) minus 119877G(120572 120573)|(1 minus


Define

D = (120572 120573) | 120588 (120572 120573) lt 1 120572 120573 isin C (23)



minus1E with 119911 = 120572 + 120573


and FΔ119905






Δ119879

and FΔ119905














isdefined by IMEXF



Figure 1(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIM-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816

RIMEX-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816


0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300


minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10


minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)



Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198882

11988621

11988622

0 sdot sdot sdot 0

1198883

11988631

11988632

11988633

0

d 0

119888119904

1198861199041

1198861199042


119886119904119904

1198871

1198872


119887119904


1198882

11988621


1198883

11988631

11988632

0 0

d 0

119888119904

1198861199041

1198861199042


0

1

2


119904

(4)


119888119894=

119894

sum

119895=1

119886119894119895=

119894minus1

sum

119895=1

119886119894119895 119894 = 2 119904 (5)




1199101015840

(119905) = 119891119904(119905 119910 (119905)) + 119891

119899119904(119905 119910 (119905)) 119905 isin [0 119879]

119910 (0) = 1199100

(6)



119870119894= 119910119899+ ℎ

119894

sum

119895=1

119886119894119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119894minus1

sum

119895=1

119886119894119895119891119899119904(119905119899+ 119888119895ℎ119870119895) 119894 = 1 2 119904

119910119899+1

= 119910119899+ ℎ

119904

sum

119895=1

119887119895119891119904(119905119899+ 119888119895ℎ119870119895)

+ ℎ

119904

sum

119895=1

119895119891119899119904(119905119899+ 119888119895ℎ119870119895)

(7)

where sum119894119895=1




0 0 0

1 0 1

0 1

0 0 0

1 1 0

1 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX Euler

(8)


0 0 0

11

2

1

2

1

2

1

2

0 0 0

1 1 0

1

2

1

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX trapezoidal

(9)



0 0 0 0 0

119886 0 119886 0 0

07179332607 0 02820667392 119886 0

1 0 12084966490 minus0644363171 119886

0 12084966490 minus0644363171 119886

0 0 0 0

119886 0 0 0

03212788860 03966543747 0 0

minus01058858296 119887 119887 0

0 12084966490 minus0644363171 119886⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

IMEX(443)

(10)





1199101015840

(119905) = 120582119910 + 120583119910 120582 120583 isin C (11)


119870 = E119910119899+ ℎ120582119860119870 + ℎ120583119860119870

119910119899+1

= 119910119899+ ℎ120582b119870 + ℎ120583b119870

(12)


119894119895)119860 =

(119860119894119895) b = (119887

1 1198872 119887

119904) and b = (

1 2

119904) Moreover

E = (1 1 1)119879


119910119899+1

= 119877 (120572 120573) 119910119899 (13)



E (14)




M119894119895=

0 if 119894 le 119895

120590119894minus119895minus1

if 119894 gt 119895(15)




10038171003817100381710038171003817M119896

(120590)10038171003817100381710038171003817infin

le


1 minus |120590|)

119896

(119873 minus 1

119896) 119894119891 |120590| lt 1

|120590|119873minus119896minus1

(119873 minus 1

119896) 119894119891 |120590| ge 1

(16)


Δ119879

and FΔ119905


GΔ119879

(119879119899 119884119896

119899 Δ119879) = 119877G (120572 120573) 119884

119896

119899

FΔ119905(119879119899 119884119896

119899 Δ119879) = 119877

119872

F (120572

119872120573

119872)119884119896

119899 119896 = 1 2

(17)



= 119899Δ119879 for 119899 =




sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 119884119896

119899

10038161003816100381610038161003816

le (

10038161003816100381610038161003816119877119872

F (120572119872 120573119872) minus 119877G (120572 120573)10038161003816100381610038161003816

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 1198840

119899

10038161003816100381610038161003816

119896 ge 1

(18)



119899= 119910(119905

119899) minus 119884119896

119899


119890119896+1

119899+1= 119877G (120572 120573) 119890

119896+1

119899+ [119877F (

120572

119872120573

119872) minus 119877G (120572 120573)] 119890

119896

119899

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]

119899

sum

119895=1

119877119899minus119895

G(120572 120573) 119890

119896

119895

(19)


119864119896= (119890119896

1 119890119896

2 119890

119896


matrix form as

119864119896+1

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]M (119877G (120572 120573)) 119864

119896

(20)


119864119896= [119877F (

120572

119872120573

119872) minus 119877G(120572 120573)]

119896

M119896(119877G (120572 120573)) 119864

0 (21)



1003817100381710038171003817100381711986411989610038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times10038171003817100381710038171003817M119896(119877G (120572 120573))

10038171003817100381710038171003817

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times (1 minus

1003816100381610038161003816119877G (120572 120573)1003816100381610038161003816119873minus1

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

(22)



The factor 120588(120572 120573) = |119877119872

F (120572119872 120573119872) minus 119877G(120572 120573)|(1 minus


Define

D = (120572 120573) | 120588 (120572 120573) lt 1 120572 120573 isin C (23)



minus1E with 119911 = 120572 + 120573


and FΔ119905






Δ119879

and FΔ119905














isdefined by IMEXF



Figure 1(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIM-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816

RIMEX-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816


0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300


minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10


minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)



Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199101015840

(119905) = 120582119910 + 120583119910 120582 120583 isin C (11)


119870 = E119910119899+ ℎ120582119860119870 + ℎ120583119860119870

119910119899+1

= 119910119899+ ℎ120582b119870 + ℎ120583b119870

(12)


119894119895)119860 =

(119860119894119895) b = (119887

1 1198872 119887

119904) and b = (

1 2

119904) Moreover

E = (1 1 1)119879


119910119899+1

= 119877 (120572 120573) 119910119899 (13)



E (14)




M119894119895=

0 if 119894 le 119895

120590119894minus119895minus1

if 119894 gt 119895(15)




10038171003817100381710038171003817M119896

(120590)10038171003817100381710038171003817infin

le


1 minus |120590|)

119896

(119873 minus 1

119896) 119894119891 |120590| lt 1

|120590|119873minus119896minus1

(119873 minus 1

119896) 119894119891 |120590| ge 1

(16)


Δ119879

and FΔ119905


GΔ119879

(119879119899 119884119896

119899 Δ119879) = 119877G (120572 120573) 119884

119896

119899

FΔ119905(119879119899 119884119896

119899 Δ119879) = 119877

119872

F (120572

119872120573

119872)119884119896

119899 119896 = 1 2

(17)



= 119899Δ119879 for 119899 =




sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 119884119896

119899

10038161003816100381610038161003816

le (

10038161003816100381610038161003816119877119872

F (120572119872 120573119872) minus 119877G (120572 120573)10038161003816100381610038161003816

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

sup119899gt0

10038161003816100381610038161003816119910 (119905119899) minus 1198840

119899

10038161003816100381610038161003816

119896 ge 1

(18)



119899= 119910(119905

119899) minus 119884119896

119899


119890119896+1

119899+1= 119877G (120572 120573) 119890

119896+1

119899+ [119877F (

120572

119872120573

119872) minus 119877G (120572 120573)] 119890

119896

119899

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]

119899

sum

119895=1

119877119899minus119895

G(120572 120573) 119890

119896

119895

(19)


119864119896= (119890119896

1 119890119896

2 119890

119896


matrix form as

119864119896+1

= [119877F (120572

119872120573

119872) minus 119877G (120572 120573)]M (119877G (120572 120573)) 119864

119896

(20)


119864119896= [119877F (

120572

119872120573

119872) minus 119877G(120572 120573)]

119896

M119896(119877G (120572 120573)) 119864

0 (21)



1003817100381710038171003817100381711986411989610038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times10038171003817100381710038171003817M119896(119877G (120572 120573))

10038171003817100381710038171003817

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times (1 minus

1003816100381610038161003816119877G (120572 120573)1003816100381610038161003816119873minus1

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

(22)



The factor 120588(120572 120573) = |119877119872

F (120572119872 120573119872) minus 119877G(120572 120573)|(1 minus


Define

D = (120572 120573) | 120588 (120572 120573) lt 1 120572 120573 isin C (23)



minus1E with 119911 = 120572 + 120573


and FΔ119905






Δ119879

and FΔ119905














isdefined by IMEXF



Figure 1(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIM-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816

RIMEX-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816


0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300


minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10


minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)



Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817100381711986411989610038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times10038171003817100381710038171003817M119896(119877G (120572 120573))

10038171003817100381710038171003817

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

le

10038161003816100381610038161003816100381610038161003816119877F (

120572

119872120573

119872) minus 119877G (120572 120573)

10038161003816100381610038161003816100381610038161003816

119896

times (1 minus

1003816100381610038161003816119877G (120572 120573)1003816100381610038161003816119873minus1

1 minus1003816100381610038161003816119877G (120572 120573)

1003816100381610038161003816

)

119896

10038171003817100381710038171003817119864010038171003817100381710038171003817infin

(22)



The factor 120588(120572 120573) = |119877119872

F (120572119872 120573119872) minus 119877G(120572 120573)|(1 minus


Define

D = (120572 120573) | 120588 (120572 120573) lt 1 120572 120573 isin C (23)



minus1E with 119911 = 120572 + 120573


and FΔ119905






Δ119879

and FΔ119905














isdefined by IMEXF



Figure 1(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161 (119909 + 119894119910 minus 1) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIM-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus10038161003816100381610038161 (119909 + 119894119910 minus 1)

1003816100381610038161003816

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816(1 + 119894119910) (1 minus 119909) minus (1 + (119909 + 119894119910)119872)

11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816

RIMEX-IMEX (119909 119910)

=


11987210038161003816100381610038161003816

1 minus1003816100381610038161003816(1 + 119894119910) (1 minus 119909)

1003816100381610038161003816


0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300


minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10


minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)



Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 00

20

40

60

0

100

200

300

400

0

10

20

30

40

0

100

200

300


minus60

minus40

minus20

minus400

minus300

minus200

minus100

minus40

minus30

minus20

minus10

minus400

minus300

minus200

minus100

minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(a)

0

20

40

60

80

100

0

20

40

60

80

100

0

10

20

30

40

50

0

20

40

60

80


minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus80

minus60

minus40

minus20

minus40

minus30

minus20

minus10


minus1000

minus500

minus100

minus50 0 0

minus1000

minus500

minus100

minus50

(b)

0

0

50

100

1500

0

200

400

600

0

20

40

60

0

200

400


minus100

minus200

minus150

minus100

minus50

minus600

minus400

minus200

minus4000

minus2000 0 0

minus100

minus200

minus4000

minus2000

minus60

minus40

minus20

minus600

minus400

minus200

(c)



Figure 1(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1minus (1 +

(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1

119909 + 119894119910 minus 1

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

119872100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 119894119910


(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

1 + 119894119910

1 minus 119909

10038161003816100381610038161003816100381610038161003816)

minus1

(24)


0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

1

2


minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(a)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(b)


0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

0

0

1

2

3

minus1

minus5

minus2

minus3

(c)






Figure 2(a)

RIM-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1


10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

20

00204060810

02

04

06

08

1

12

14

Time tSpace x

U(tx)

(a)

10

20

00204060810

05

1

15

2

25

3

35

Time t

Space x

V(tx)

(b)


RIM-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119872 + 119894119910

119872 minus 119909)

119872100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

Figure 2(b)

RIM-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIM-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910


(2 + 119894(119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

10038161003816100381610038161003816100381610038161003816

2 + 119909 + 119894119910

2 minus 119909 minus 119894119910

10038161003816100381610038161003816100381610038161003816)

minus1

RIMEX-EX (119909 119910)

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)


119909 + 119894119910

119872+(119909 + 119894119910)

2

21198722)

11987210038161003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

RIMEX-IMEX (119909 119910)

=

1003816100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

minus(1 +(2 + 119894 (119910119872)) (119909 + 119894119910)

2119872 minus 119909)

1198721003816100381610038161003816100381610038161003816100381610038161003816

times (1 minus

100381610038161003816100381610038161003816100381610038161003816

1 +(2 + 119894119910) (119909 + 119894119910)

2 minus 119909

100381610038161003816100381610038161003816100381610038161003816

)

minus1

(25)



0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

05

1

0

05

1

15

0

05

1

15

0

05

1

15

2

0

1

2

3

0

1

2

3

4

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

T

X

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

00

05

1

10

20

TX

U V U

U U

U U

V

V V

V V

times10minus3


minus4times10

minus4

times10minus5

times10minus6

times10minus6

times10minus5

times10minus6

times10minus7


minus7





Δ119879



5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30

10minus10

10minus5

100

105


(a)

0

0

02

04

06

0

0

10

20

30

40

0

0

10

20

30

40

50

minus2 minus1

minus06

minus04

minus02


minus30

minus20

minus10


minus40

minus30

minus20

minus10


(b)



4 Numerical Results


119906119905= 1205981119906119909119909

minus 119906V2 + 120578 (1 minus 119906)

V119905= 1205982V119909119909

+ 119906V2 minus (120578 + 120579) V(26)



119906 (0 119909) = 1 minus1

2(sin (3120587119909))100

V (0 119909) =1

4(sin (3120587119909))100

119906 (119905 1) = 119906 (119905 0) = 1

V (119905 1) = V (119905 0) = 0

(27)



asymp

(119906119895+1

minus2119906119895+119906119895minus1

)Δ1199092 and V

119909119909asymp (V119895+1


)Δ1199092 with






Δ119879and F

Δ119905being





Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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