High order explicit methods for parabolic equationsHigh order explicit methods for parabolic equationsand stiff ODEsand stiff ODEs

(DUMKA3, DUMKA4, ROCK2, ROCK4)(DUMKA3, DUMKA4, ROCK2, ROCK4)

Alexei MedovikovTulane University

A. Abdulle, A. Medovikov Second order Chebyshev methods based on orthogonal polynomials. Numerische Mathematik V90.1. pp.1-18 Medovikov A.A. High order explicit methods for parabolic equations. BIT, V38,No2,pp.372-390Lebedev V.I., Medovikov A.A. Method of second order accuracy with variable time steps. Izv. Vyssh. Uchebn. Zaved. Mat. no. 9, 52--60 (English translation).

WELCOME TO DUMKALANDExplicit numerical methods for stiff differential equations

DUMKA3 - integrates initial value problems for systems of first order ordinary differential equations y'=f(y,t). It is based on a family of explicit Runge-Kutta-Chebyshev formulas of order three. It uses optimal third order accuracy stability polynomials with the largest stability region along the negative real axis.

  Dumka3 Examples

  Download DUMKA3.cpp (C++)

  Download DUMKA3.c (C)

  Download DUMKA3.f (FORTRAN)

  Download ROCK2/ROCK4 (rock.tar)(FORTRAN)

  Applications (medicine, biology, apply math ...)

  Motility of microorganisms

  E-Mail:amedovik@math.tulane.edu

  Phone:(504)862-8396

  Address:  Mathematics Department   Tulane University  New Orleans, LA Web: http://www.math.tulane.edu/~amedovik

Examples of solution of stiff differential equations by explicit methods

Brusselator equation

Nagumo nerve conduction equation

Burgers equation

2

2

2

22

2

2

2

22

yv

xv

vuu4.3tv

)t,y,x(fyu

xu

u4.4vu1tu

else

1.1tand1.0)6.0y()3.0x(if,0

5)u(f

)bvu(ntv

v)u(fxu

tu

222

2

2

21kk1k

21k

21kk

xuu2u

x4uu

dtdu

SummarySummary

1. Stability: Explicit methods have small stepsize , due to conditional stability

2. Variable steps can be used to maximize mean stepsize of a sequence of explicit methods

3. Optimal sequence of explicit steps can be found in terms of roots of stability polynomials, which approximate exponential function and possess Chebyshev alternation

4. Asymptotic formulas and orthogonal polynomials can be used to construct such polynomials, even very high degree polynomials (n > 1000)

5. Accuracy: In order to construct high order explicit methods for non-linear ODE, we start with stability polynomials and we use B-series in order to satisfy order conditions, and build Runge-Kutta methods for non-liner ODEs

6. Efficient stepsize control and step rejection procedure are achieved via embedded methods

7. For automatic computation of spectral radius we used non-linear power method.

)n(n/ncn/ 2p

n

i

n

1p

ii

pn za!p/zz1)z(R

00 y)t(y)t,y(fdtdy

λyy'

0001 y)1(yyy

0n

n y)1(y

)t,y(fyy

0001

Explicit Euler method:

Stability analysis of explicit RK methods

ODEs:

Test equation:

Stability function:

Stability region: hz,1)z(R:CzS n

Goal:

}S)y/f(Sp{ Find stability polynomial which maximize average stepsize , given

h0nn y)h(Ry is a total stepwhere

n/h

M/2

01 y)1(y

11)(Rn

000001 err))y(JI(err)y(Jerrerr )y~y)(y(J)y~y(

)y~(fy~())y(fy(y~y

00000

000011

Stability analysis of explicit RK methods

/2

Explicit Euler method:

Stability condition of explicit Euler method:

h

Linear stability analysis for non-linear ODEs:

000 erryy~

where )y/)y(f(M

Linear stability RK methods vs. Stability RK methods?

})y/f(Sp{ regionstability

z,h,z1)z(R

)1()h(R i

n

1in

0

nn

1ii d

)(dRh

Can we solve stiff ordinary differential equations (ODE) by explicit Can we solve stiff ordinary differential equations (ODE) by explicit methods with stepsize larger than 2/M?methods with stepsize larger than 2/M?

Example:Example:

500/2

0)0())cos((500'

ytyy

1 2Consider two steps and where 500/11

)cos()5001(

)cos()5001(

102122

01011

tyy

tyy

)cos()cos()5001()5001)(5001( 1020120122 ttyy

yinstabilitif 500/221 stabilityif 500/112

2anyfor 10)5001)(5001( 12

h),1()h(R1

)hh(h),h1)(h1()h(R 21211

n

1iii

n

1i1 hh),h1()h(R

))tcos(y(500'y

Original idea of Runge-Kutta-Chebyshev methodsOriginal idea of Runge-Kutta-Chebyshev methods

Consider sequence of Euler steps and find an optimal polynomial

n

1i 0

ni

n

1iin

n

1ii

n

1i0in

d)(dR

n/h

S]M,0[)A(Sp1|)1(||)(R|

hy)1(y

as large as possible

ii /1

If we have found the optimal stability polynomial, the variable sequence of steps can be found in terms of the roots of the stability polynomial

The solution for n-stage Runge-Kutta-Chebyshev method order p=1 is given by Chebyshev stability polynomial.

1x,

M

x1xx,

xarccosncosxarccosncos

)(R 00

00

n

euler2

20

1x

0

02

0

0

0

n0

nhM2

nn/M2

nh

M2

n)x(hlim)1(h

Mx1

))xarccos(ncos(x1

))xarccos(nsin(n

d)(dR

)x(h

0

00 x)0(x,x)(x1M0

MMx1

1x,1)(Rmax

1xarccosncos

1

0

0n

0

Runge-Kutta methodsRunge-Kutta methods

n

1jjj0j01

n

1jjj0ij0i

).Y,hct(fbhyy

),Y,hct(fahyY

n1n21

nn1nn2n1n

n1n12n11n

n221

n11n11211

n

1n

2

1

bbbbaaaa

aaa

aa

aaaa

c

c

c

c

b

Ac

y'y

hz

zyy

z

)y,,y(,Y)(bhyy

)Y,,Y(,Y)(ahyY

t

01

t00

n

1jjj01

tn1

n

1jjij0i

Yb

YΑyY

y

Y

0

0

n

1pi

ii

p2

1ntnt2t

n

ijn

t0

1t

1

zd!p/z!2/zz1

zzz1)z(R

,ij,0a,

)11(,y))z(z1(y

1b1b1b

0A

11Ib

if

Stability function of explicit Runge-Kutta method

Theorem (T. Riha): Among all polynomials of the order p the polynomial which possess Chebyshev alternant, would maximize real stability interval

n

1pi

ii

p2n za!p/z2/zz1)z(R

]0,l[z1)z(R nn for

zlM

n

n

1pi

in

i

pn

2nn

n

Ml

d!p/Ml

!2/Ml

Ml

1)(R

or equivalently, the polynomial which possess Chebyshev alternant:

has maximal possible stepsize

M

l

d

)(dRh nn

, given stability ]M,0[,1)(Rn

nl

dzz1x1

xz)x(ln)z(ln

21

)(where

)xarccos(),x())(ncos(2

)x(P)x(

asformalasymptoticinexpressedis]1,1[intrvalthein)x(functionweight

thewithzerofromdeviationleasttheof)x(PpolynomialtheThen

).numberspositivefixedareL,,cwhere(L|ln||)x()x(|andc)x(c0

satisfieswhich]1,1[intervaltheonfunctionweightpositiveabe)x(wLet:Theorem

2

21

1

nn

2/1

n

2,11

21

dzz1x1

xz)x(wln)z(wln

21

)(where

)xarccos(),)n(ln())(ncos()x(P)x(w

:]1,1[onuniformlysatisfy),x1(sqrt/)x(w

functionweightthewithassociated)x(PspolynomialorthogonaltheThen

).numberspositivefixedareL,,cwhere(L|ln||)x(w)x(w|

andc)x(wc0satisfieswhich]1,1[intervaltheon

functionweightpositiveabe)x(wLet:Theorem

2

21

1

n

2

n

2,11

21

)1(

l)0(Rl)0(Rl)0(Rl)0(R1)0(R4p

l)0(Rl)0(Rl)0(R1)0(R3p

l)0(Rl)0(R1)0(R2p

l)0(R1)0(R1p

4n

''''n

3n

'''n

2n

''nn

'nn

3n

'''n

2n

''nn

'nn

2n

''nn

'nn

n'nn

Two algorithms of computation of stability polynomials:

1. For given n calculate weight

and roots via asymptotic formula for polynomials

of the least deviation from zero:

so that the polynomial satisfies (1),

2. For given n calculate weight

so that the polynomial satisfies (1),

where is orthogonal polynomial with the weight

)(P)(w)(R pnpn

p

1jjp )xx()x(w

)(xP pn 2

2

x1

))x(w(

)(R)()(R pnpn

)cos(x,pn)(

pn)5.0k(

pn,,1k),5.0k()()pn(

kk

jk1j

k

kk

p

1jjp )xx()x(

)(xP pn

DU

MK

A3,

4R

OC

K2,4,

RK

C

)x(T)x(T)x(T)x(T)x(T

)x(T)x(T)x(T)x(T)x(T

)x(T)x(T)x(T)x(T)x(T

)x(T)x(T)x(T)x(T)x(T

)x(T)x(T)x(T)x(T)x(T

)x(wC

)x(P

4k2'

4k1'

4k24k14k

3k2'

3k1'

3k23k13k

2k2'

2k1'

2k22k12k

1k2'

1k1'

1k21k11k

k2'k1

'k2k1k

22

k

kl2

21

1

lk dxx1

)x(w)x(P)x(P

]0,M[z),z(P)z(P)z()z(P 2kk1kkkk

2

1jj2 )xx()x(w

))(ncos()x(P)x(w)x(R 2n2n

2n

''nn

'nn

n'nn

l)0(Rl)0(R1)0(R2p

l)0(R1)0(R1p

Accuracy:Accuracy:Order conditions of Runge-Kutta methodsOrder conditions of Runge-Kutta methods

KIK

JI

KIJIK

3IJ

I

2JJ

0

3

3

J32

2

J2JJ0

J

ffffff6h

ff2h

hfy

6h

dtyd

2h

dtyd

hdtdy

y)h(y

KIK

JI32

KIJIK31

2IJ

I21JJ

0J ffftbffftb

6h

fftb2h

hf)(by)h(y~

Taylor expansions of the exact solution and numerical solution :

1Α

1Α

1Α

1

3t

n

1l,k,j,ijljkiji44

n

1l,k,j,ijljkiji43

n

1l,k,j,ijlikiji42

n

1l,k,j,iilikiji41

2tn

1k,j,ijkiji31

n

1k,j,iikiji31

t

ij

n

1j,ii21

tn

1ii

b24

aaab24)t(baaab12)t(baaab8)t(baaab4)t(b

b6aab6)t(baab3)t(b

b2ab2)t(b

bb)(b

KK

I

I

JKI

KJI

KIKI

J2KIJ

IKI

I

JIJ

I fyf

yf

fff,ffyyf

fff,fyf

ff

where

Construction of Construction of ppthth order composition method order composition method

Let us consider two consecutive steps by Runge-Kutta methods A and B. We call the method which is the result of one step of A and one step of B as the composition method C=B(A)

Stability function of the composition method C is the product of stability functions of the methods A and B

Theory of composition methods allows to calculate Taylor expansion of composition methods:

)()()(4)()(6)()(4)()(

)()()(4)()(6)()(4)()()()()(3/1)()(3/24

)()(3/2)()(3/16)()()(4)()(

)()()(4)()(6)()(4)()(

)()()(3)()(3)()(

)()()(3)()(3)()(

)()()(2)()(

)()()(

44322121324444

43312

21314343

423231

22121212142

42

41312

213

4141

3221213232

312

21212

3131

212121

taatbtatbtabtbtc

taatbatbtabtbtctaatbatb

atbtatbatabtbtc

taatbatbabtbtc

taatbtabtbtc

taatbtabtbtc

taabtbtc

abc

Given method A, define method B such that method C=B(A) will be Given method A, define method B such that method C=B(A) will be method of the order p and stability function of the method C will be method of the order p and stability function of the method C will be product of the stability functions of the methods A and B.product of the stability functions of the methods A and B.

Coefficients of Taylor expansion of the method B can be expressed in terms of coefficients of the methods C and B

))t(a)(a)t(b4)t(a)t(b6)t(a)(b4(1)t(b

))t(a)(a)t(b4)(a)t(b6)t(a)(b4(1)t(b))t(a)(a)t(b3/1)(a)t(b3/24

)(a)t(b3/2)t(a)t(b3/16)(a)t(a)(b4(1)t(b

))t(a)(a)t(b4)(a)t(b6)(a)(b4(1)t(b

))t(a)(a)t(b3)t(a)(b3(1)t(b

))t(a)(a)t(b3)t(a)(b3(1)t(b

))t(a)(a)(b2(1)t(b

)(a1)(b

443221213244

43312

213143

423231

221212121

42

41312

213

41

32212132

312

21212

31

2121

1)t(c,1)t(c

,1)t(c,1)t(c1)t(c,1)t(c1)t(c1)(c

4p3p2p1p

4443

4241

313121

pn,,2j,YY)Y(fhY

)Y(fhyY

yY

2jj1jj1jjj

0101

00

pn,,1i),t,y(fyy 00i1i

Examples of the method A (RKC) and DUMKA:

Equations for coefficients of the method BEquations for coefficients of the method B

24/)t(bcaab

6/)t(b)caca(bcab

8/)t(b)caca(cbccab

6/)t(b)caca(bcab

4/)t(bcbcbcb

3/)t(bcbcbcb

2/)t(bcbcbcb

)(bbbbb

4p

6/)t(bcab

3/)t(bcbcb

2/)t(bcbcb

)(bbbb

3p

2/)t(bcbcb

)(bbbb

2p

44242434

43

2343

22424

22323

423432424432323

3234324242323

41

344

333

322

31

244

233

222

21443322

3321

322322

31

233

222

213322

321

213322

321

Embedded methodsEmbedded methods

)h(Cdh

yd)!1p(

hy)h(y

)h(Cdh

yd)!1p(

hyy

)h(Cdhyd

!ph

yy

2p21p

1p1p

0

2p11p

1p1p

01p

1

1p0p

pp

0p1

)h(OhCdh

yd)!1p(

1yyerr

)h(OhCdh

yd)!1p(

1y)h(yerr

2p1p01p

1pp1

1p1

2p1p01p

1pp1

1p/1

oldnew

1pnew

1pold

1poldnew

oldnew

new

1pnewnew

p1new

errtol

hh

tolhh

err

h/errC

CC

tolerr

hCy)h(yerr

Embedded methodsEmbedded methods

2/1cb~1b~

and

1)t(b,1)t(b

2/1cb

1b

i

i

3231

ii

i

11

n

1jjj0j01

n

1jjj0j01

n

1jjj0ij0i

y~yerr

),Y,hct(fb~hyy~

),Y,hct(fbhyy

),Y,hct(fahyY

Embedded composition method C’=B’(A)Embedded composition method C’=B’(A)

4321

111

3231

21

3

2

b~b~b~b~bbb

0aa

00a

000

c

c

0

2/)t(b)(bb~cb~cb~)(bb~b~b~b~

2142322

4321

6/)t(b)cbcbcb(b~)caca(b~cab~3/)t(b)(bb~cb~cb~cb~

2/)t(b)(bb~cb~cb~cb~

)(bb~

32433322534324242323

312

5244

233

222

215443322

5

1ii

54321

4321

434241

3231

21

4

3

2

b~b~b~b~b~bbbb

0aaa

00aa

000a

0000

c

c

c

0

Numerical resultsNumerical results

]5.2,0[t],1,0[x

,0005.0,)x1(x5.1)0(ux

uu2ux4uu

dtdu

2

21kk1k

21k

21kk

|)(|max]0,1[1|)(| inn RR

)2()1(,)()(

)()()( 00

00

xxxxPxw

xPxwR

pnp

pnpn

)y(fdtdy