A Dispersion Minimizing Finite Difference Scheme and Precond - Chen Et Al - 2012

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A dispersion minimizing finite difference scheme and preconditioned

solver for the 3D Helmholtz equationq

Zhongying Chen a,1, Dongsheng Cheng a,, Tingting Wu b

a Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, PR Chinab School of Mathematical Sciences, Shandong Normal University, Jinan 250014, PR China

a r t i c l e i n f o

Article history:

Received 29 November 2011Received in revised form 27 July 2012Accepted 30 July 2012Available online 17 August 2012

Keywords:

Helmholtz equationPerfectly matched layerFinite difference methodPreconditionerBi-CGSTABShifted-LaplacianMultigridProlongation operator

a b s t r a c t

In this paper, a new27-point finite difference method is presented for solving the3D Helm-holtz equation with perfectly matched layer (PML), which is a second order scheme andpointwise consistent with the equation. An error analysis is made between the numericalwavenumber and the exact wavenumber, and a refined choice strategy based on minimiz-ing the numerical dispersion is proposed for choosing weight parameters. A full-coarseningmultigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear sys-tem stemming from the Helmholtz equation with PML by the finite difference scheme. Theshifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectralanalysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method,with a multigrid method used to invert the preconditioner approximately. Full-coarseningmultigrid is employed, and a new matrix-based prolongation operator is constructedaccordingly. Numerical results are presented to demonstrate the efficiency of both the

new 27-point finite difference scheme with refined parameters, and the preconditionedBi-CGSTAB method with the 3D full-coarsening multigrid.2012 Elsevier Inc. All rights reserved.

1. Introduction

The wave equation has numerous important applications in sciences and engineering, for instance, in geophysics, aero-nautics, marine technology. Applying the Fourier transform with respect to time to the wave equation, we obtain the fre-quency domain wave equation, which is the well-known Helmholtz equation. The Helmholtz equation is so importantthat its numerical simulation has stimulated significant research. To solve the Helmholtz equation numerically, artificialboundary conditions are often employed so that we can truncate the infinite computing domain into a finite one. The per-fectly matched layer (PML, cf.[8,30,40]) proposed by Brenger is a popular artificial absorbing boundary condition, which is

used to gradually damp the outgoing waves and eliminate boundary reflections. For convenience, we call the Helmholtzequation with PML the Helmholtz-PML equation, which is considered in this paper.

To discretize the Helmholtz equation, we mainly have finite difference methods (cf.[11,19,22,30,31,33,34,43]) and finiteelement methods (cf.[2,3,10,13,16,20]). Finite difference methods are commonly used in engineering field such as geophys-ics. In scientific computing, solving the Helmholtz equation numerically with high wavenumbers still remains as one of themost difficult tasks. Due to the pollution effect of high wavenumbers, the wavenumber of the numerical solution is different

0021-9991/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jcp.2012.07.048

q This research is partially supported by the Guangdong Provincial Government of China through the Computational Science Innovative Research Teamprogram. Corresponding author.

E-mail addresses:[email protected](Z. Chen),[email protected](D. Cheng),[email protected](T. Wu).1 Supported in part by the Natural Science Foundation of China under Grants 10771224 and 11071264.

Journal of Computational Physics 231 (2012) 81528175

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from the wavenumber of the exact solution, which is known as numerical dispersion (cf.[20,21]). The conventional 2D5-point and 3D 7-point finite difference schemes lead to serious numerical dispersion, polluting the numerical accuracy.To reduce the numerical dispersion, some weighted transformed-based finite difference schemes have been constructed(cf. [19,22,24,25,28,32]), which need less grids per wavelength, while maintaining a comparable accuracy. For the 3DHelmholtz equation, the weighted transformed-based finite difference schemes are very complicated. For instance, in[25], seven rotated coordinate systems were employed to construct a 3D 27-point difference scheme.

In this paper, we shall propose an alternative finite difference scheme for the 3D Helmholtz equation, which is an exten-sion of our previous work for the 2D case in[11]. This new scheme remains weighted, but is rotation-free. We call it a dis-persion minimizing finite difference scheme, since the weight parameters are obtained by minimizing the numericaldispersion. The dispersion minimizing finite difference scheme is of second order and pointwise consistent. Its constructionis much simple without rotating the coordinate system in 3D space, compared with the staggered-grid 27-point formulation,which was originally proposed for the wave equation in [24], and was further developed to the 3D Helmholtz equation in[25]. Interestingly, we shall present that our scheme is equivalent to the scheme in [25]under certain conditions. Moreover,weight parameters of our scheme are chosen by refining parameter intervals, which is called as the refined choice strategy.We shall give an error analysis between the numerical wavenumber and exact wavenumber, and numerical experimentsshow that the new scheme with the refined strategy outperforms the staggered-grid scheme in reducing the numericaldispersion.

For the Helmholtz equation, high-order finite difference schemes (cf. [4,31,34]) are also constructed to improve thenumerical accuracy. For instance, in [34], compact finite difference schemes of sixth order are proposed for the 3D Helmholtzequation. These sixth order schemes perform pretty well for small constant wavenumbers. Theoretically, sixth orderschemes are more competitive, so long as the step size is small enough. However, grids per wavelength can not be too muchin practical, that is, the step size can not be too small. Also, the pollution analysis of error shows that the accuracy not onlydepends on the convergence order, but also the wavenumber. Then, though the sixth order scheme has a higher convergenceorder, it does not always means a higher accuracy. For certain step size and large wavenumbers, the new second orderscheme may compete with the sixth order scheme, since it minimizes the numerical dispersion. In this paper, we shall com-pare the new second order scheme with the 3D sixth order compact scheme in[34], and numerical examples show that thenew second order scheme performs better for certain step sizes and large wavenumber. Moreover, we specially point outthat sixth order schemes are more demanding, since they require the solution and source term be continuously differentiableof sixth and fourth order, respectively. They also require the wavenumber be constant and the step sizes be equal in threedirections. However, these requirements may not be met in practice. For example, in geophysical applications, we have todeal with the Helmholtz equation with varying wavenumbers in heterogeneous medium, and the step size in the third direc-tion may differ from others. In addition, high order schemes may have difficulties in dealing with boundary conditions, and ahigh convergence order may not be obtained if the boundary condition is not dealt properly.

After discretization of the Helmholtz equation, the preconditioned Bi-CGSTAB method is employed to solve the largeindefinite linear system, and the shifted-Laplacian (cf.[14,15,23]) is considered as the preconditioner. The shifted-Laplacianpreconditioner is an extension of the Laplacian preconditioner, which was originally proposed in [5,6] for the 2D case. In thispaper, for the 3D Helmholtz-PML equation, the corresponding preconditioner we employ is the 3D complex shifted-Lapla-cian-PML. We specially analyze the spectral distribution of the linear system from the perspective of linear fractal mappingin complex variable functions. We propose a new prolongation operator for the 3D full-coarsening multigrid, which is usedto invert the preconditioner approximately. With the same number of iterations, it is expected that the full-coarsening mul-tigrid shall consume less CPU time than the semi-coarsening case, which decreases more gradually in grid size. Numericalresults are presented to illustrate that the 3D full-coarsening multigrid with the new prolongation operator gives a betterperformance, reducing both the number of iterations and the total CPU time needed for convergence. In the experiment,wavenumbers range from constant in homogeneous medium to greatly varying ones in heterogeneous medium. For the caseof constant wavenumber, the dimensionless wavenumber (cf. [21]) we compute in our experiment is as large as 220. Wespecially point out that the number of iterations scales roughly linearly with the wavenumber, which seems to be a classicalproblem for iterative solutions of the Helmholtz equation. We have not solved this problem.

In this paper, we aims at solving the 3D Helmholtz-PML equation related with geophysical applications, and focuses onboth the discretization of the operator equation and iterative method of the discrete linear system. The remainder of thispaper is organized as follows. In Section2, a new 27-point finite difference scheme is developed and analyzed. In Section3,an error analysis is presented between the numerical wavenumber and exact wavenumber. In Section 4, a refined choicestrategy is given to choose weighted parameters of the new scheme. In Section5, we discuss the 3D complex shifted-Laplacian preconditioning, and make some spectral analysis. In Section6, we propose a new prolongation operator for thefull-coarsening multigrid. In Section7, some numerical experiments are presented. Finally, in Section 8, some conclusionsare drawn.

2. A consistent 27-point finite difference scheme for the 3D Helmholtz-PML equation

In this section, we formulate a new 27-point finite difference scheme for the 3D Helmholtz-PML equation, based on theidea of weighted average (cf. [30]). This scheme is pointwise consistent with the equation and of second order. Compared

Z. Chen et al. / Journal of Computational Physics 231 (2012) 81528175 8153
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with methods in[24,25], the construction of this scheme is much simpler and can be easily extended to nonuniform grids,since it neither needs the first-order hyperbolic system nor the rotated Cartesian coordinate system.

We consider the 3D Helmholtz equation for wave problems

Au: Du 1 aik2u g in R3; 2:1

where D: @2@x2

@2@y2

@2@z2

is the 3D Laplacian, unknown u usually represents a pressure field in the frequency domain,

k:

2pf=vis the wavenumber withf indicating the frequency in Hertz and vindicating the wave velocity,ais the real num-

ber indicating the fraction of damping in the medium, iffiffiffiffiffiffiffi1p is the imaginary unit, andgrepresents the source term. The

wavenumber k, which depends explicitly on the spatial velocity v, is a constant for the homogeneous medium and a variablefor the heterogeneous medium. The medium is considered to be barely attenuative when 0 6 a1, andacan be set up to5% (i:e:;a0:05 in geophysical applications. When a square domain of size His normalized to a unit domain, we obtain thedimensionless wavenumber which equals to 2pfH=v (cf.[21]). In the remainder, the wavenumber refers to the dimension-less wavenumber.

Applying the PML technique to truncate the infinite domain in(2.1)into a bounded domain, we have the 3D Helmholtz-PML equation

Au: @

@x A

@u

@x

@

@y B

@u

@y

@

@z C

@u

@z

1 aiDu g; 2:2

with

A:nyynzznxx

; B:nxxnzznyy

;

C :nxxnyy

nzz ; D:nxxnyynzzk

2x;y;z:

Here, nxx; nyyand nzzare 1D damping functions that satisfy nxx 1; nyy 1, nzz 1 in the interior area.We next introduce the construction of the new 27-point finite difference scheme. Firstly, we present the 27-point finite

difference stencil with numbering in Fig. 1, where0;0;0 represents the center point in the stencil, andl;m; n withl;m; n2 f1;0;1g denote the points surrounding 0;0;0. For convenience, 0;0;0 and l; m; n are identified withx0;y0;z0 andx0 lh;y0 mh;z0 nh respectively, where h is the discretization step. The discretization of a functionuat pointl; m;n is denoted by u l;m;n :ux0 lh;y0 mh;z0 nh. Then, to approximate @@x A @u@x

, @@y

B @u@y

and @

@z C@u

@z

, we

utilize

Lh;xu:c1Lh;xuj 0;0;0 c24 Lh;xuj 0;1;0 Lh;xuj 0;1;0 Lh;xuj 0;0;1 Lh;xuj 0;0;1 h i

c34

Lh;xuj 0;1;1 Lh;xuj 0;1;1 Lh;xuj 0;1;1 Lh;xuj 0;1;1

h i; 2:3

y z

x

(0,1,0)

(0,0,0)

(0,0,1)

(0,0,1)

(0,1,0)

(1,0,0)

(1,0,0)

Fig. 1. The 27-point finite difference stencil with numbering.

8154 Z. Chen et al. / Journal of Computational Physics 231 (2012) 81528175


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Lh;yu:c1Lh;yuj 0;0;0 c24

Lh;yuj 0;0;1 Lh;yuj 0;0;1 Lh;yuj 1;0;0 Lh;yuj 1;0;0

h i

c34

Lh;yuj 1;0;1 Lh;yuj 1;0;1 Lh;yuj 1;0;1 Lh;yuj 1;0;1

h i; 2:4

Lh;zu:c1Lh;zuj 0;0;0 c24

Lh;zuj 1;0;0 Lh;zuj 1;0;0 Lh;zuj 0;1;0 Lh;zuj 0;1;0

h i

c3

4 Lh;zu

j 1;1;0 Lh;zu

j 1;1;0 Lh;zu

j 1;1;0 Lh;zu

j 1;1;0 h i;

2:5

whereLh;xuj 0;m;n ;Lh;yuj l;0;n andLh;zuj l;m;0 (l;m; n2 f1;0;1g) are approximations of @@x A @u@x

; @@y B@u@y

and @@z C

@u@z

respec-

tively, based on the second order centered difference, and parameters satisfy c1 c2c31. The differential operatorL: @

@x A @@x @

@y B @@y

@

@z C @@z

is then approximated byLh,

Lhu: Lh;xu Lh;yu Lh;zu: 2:6

Finally, the zeroth order term Du is approximated by a weighted average

IhDu:w1D0;0;0u0;0;0w2Ih;1Du w3Ih;2Du w4Ih;3 Du ; 2:7

with parameterswjj1;2;3;4satisfyingP4

j1wj1. Here,Ih is an average operator, and operatorsIh;1;Ih;2andIh;3 aredefined by

Ih;1Du: 16 D1;0;0u1;0;0D0;1;0u0;1;0D0;0;1u0;0;1D1;0;0u1;0;0D0;1;0u0;1;0D0;0;1u0;0;1 ;

Ih;2Du: 1

12 D1;1;0u1;1;0D0;1;1u0;1;1D1;0;1u1;0;1D1;1;0u1;1;0D0;1;1u0;1;1D1;0;1u1;0;1

D1;1;0u1;1;0D0;1;1u0;1;1D1;0;1u1;0;1D1;1;0u1;1;0D0;1;1u0;1;1D1;0;1u1;0;1;

and

Ih;3Du:1

8 D1;1;1u1;1;1D1;1;1u1;1;1D1;1;1u1;1;1D1;1;1u1;1;1D1;1;1u1;1;1D1;1;1u1;1;1

D1;1;1u1;1;1D1;1;1u1;1;1:

With

Lh and

Ih , we obtain a new 27-point finite difference scheme for the 3D Helmholtz-PML Eq.(2.2):

Lhu 1 aiIh Du g0;0;0: 2:8

To analyze the consistency of the new 27-point scheme, we recall the notion for pointwise consistency[38].

Definition 2.1. Suppose that the partial differential equation under consideration isTugand the corresponding finitedifference approximation is Tl;m;nUl;m;nGl;m;n where Gl;m;n denotes whatever approximation which has been made of thesource term g. Letxl;ym;zn : x0 lDx;y0 mDy;z0 nDz. The finite difference scheme Tl;m;nUl;m;nGl;m;n is pointwiseconsistent with the partial differential equationTugatx;y;zif for any smooth function //x;y;z,

T/gjxxl;yym ;zzn Tl;m;n/xl;ym;zn Gl;m;n

! 0 2:9

as Dx;Dy;Dz!0.For the difference approximation(2.8)of the 3D Helmholtz-PML Eq.(2.2), we have the following proposition.

Proposition 2.2. The 27-point finite difference scheme(2.8)is pointwise consistent with the 3D Helmholtz-PML Eq.(2.2).

Proof. Assume thatxl 6 x< xl1; ym 6 y< ym1andzn 6 z< zn1. We recall thatP3

j1cj1 andP4

j1wj1. Then, it followsfrom(2.3)(2.5)and the Taylor theorem that

Lh;xu @

@x A

@u

@x

l1h

2 Oh

3; 2:10

Lh;yu @

@y B

@u

@y

l2h

2 Oh

3; 2:11

Lh;zu @

@z C

@u

@z l3h2

Oh3

; 2:12



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in which

l1 : 1

24

@3

@x3 A

@u

@x

@

@x A

@3u

@x3

!" #c24

@3

@y2@x A

@u

@x

@3

@z2@x A

@u

@x

" #

c34

@

@y

@

@z

2@

@x A

@u

@x

@

@y

@

@z

2@

@x A

@u

@x

" #;

l2 : 1

24

@3

@y3 B

@u

@y

@

@y B

@3u

@y3

!" #c24

@3

@x2@y B

@u

@y

@3

@z2@y B

@u

@y

" #

c34

@

@x

@

@z

2@

@y B

@u

@y

@

@x

@

@z

2@

@y B

@u

@y

" #;

and

l3 : 1

24

@3

@z3 C

@u

@z

@

@z C

@3u

@z3

!" #c24

@3

@x2@z C

@u

@z

@3

@y2@z C

@u

@z

" #

c34

@

@x

@

@y

2@

@z C

@u

@z

@

@x

@

@y

2@

@z C

@u

@z

" #:

Similarly, following(2.7)and the Taylor theorem, we obtain that

IhDu Du l4h2

Oh3

; 2:13

where

l4 : w26

w33

@2@x2

@

2

@y2

@2

@z2

!Du

w48

@

@x

@

@y

@

@z

2

@

@x

@

@y

@

@z

2

@

@x

@

@y

@

@z

2

@

@x

@

@y

@

@z

2" #Du:

It follows from Eqs.(2.10)(2.13)that the left hand side of the 27-point finite difference approximation (2.8)is equivalent to

@

@x A@u

@x

@

@y B@u

@y

@

@z C@u

@z

1 aiDuhh2h2 Oh3; 2:14

where hh2 :P4

j1lj. Because ofR3

j1cj1; R4j1xj1, and Definition2.1, we come to our conclusion. The scheme is justsecond-order accurate. h

The next proposition indicates the relationship between the new 27-point finite difference scheme (2.8)and the stag-gered-grid 27-point finite difference scheme proposed in[24,25].

Proposition 2.3. For the case of the wavenumber being a constant, the staggered-grid 27-point finite difference scheme andthe 27-point finite difference scheme(2.8)are equivalent ifwjwmjj1;2; 3;4, and

c1 cs12

3cs2

1

2cs3; c2

1

3cs2; c3

1

2cs3;

wherewmjj1;2;3;4andcsjj1;2;3are parameters in the staggered-grid 27-point finite difference scheme.

Proof. For convenience, we firstly introduce the notations

R0 :u0;0;0;

R1 :u1;0;0u0;1;0u0;0;1u1;0;0u0;1;0u0;0;1;

R2 :u1;1;0u0;1;1u1;0;1u1;1;0u0;1;1u1;0;1u1;1;0u0;1;1u1;0;1u1;1;0u0;1;1u1;0;1;

and

R3 :u1;1;1u1;1;1u1;1;1u1;1;1u1;1;1u1;1;1u1;1;1u1;1;1:

When the wavenumber is a constant, the staggered-grid 27-point scheme in[25]reduces to



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cs1h2 R1 6R0

cs23h

2

1

2R2 R1 12R0

cs34h

2

3

2R3 R2 2R1 12R0

1 aik2 wm1R0wm26 R1

wm312

R2wm48 R3

g0;0;0;

whereP4

j1wmj1 andP3

j1csj 1. For the details of the scheme, we refer readers to the derivation of the Eq. (6) in[25]. Forour proposed 27-point finite difference scheme, the formula(2.8)reduces to

c1h2 R1 6R0 c2

h2 12

R2 R1

c3h2 34

R3 12R2

1 aik2 w1R0w2

6 R1w3

12R2w4

8 R3

g0;0;0: 2:16

For Eqs.(2.15)and(2.16), comparing parameters ofR0, R1;R2 and R3 yields the conclusion of this proposition. h

3. Error analysis between the numerical wavenumber and the exact wavenumber

In this section, a classical dispersion analysis is made to assess the accuracy of the new scheme(2.8), and a theoreticalresult is given on the approximation of numerical wavenumber to the exact wavenumber when kh is small enough.

To do a dispersion analysis for the new scheme(2.8), we consider an infinite homogeneous model with constant velocityv, and we also assume that the medium has no damping effect on the waves, that is,a0 for the Eq.(2.1). Thus, we obtainthe Eq. (2.16) with a0. Let k be the wavelength, and G: khbe the number of gridpoints per wavelength. Since k vfand thewavenumber k:

2pf

v, we have kh

2p

G

. Following the classical harmonic approach, we firstly insert the discrete expression of

a plane waveu lmneihk l cos/ coshm cos/ sinhn sin/ in(2.16), where p2 /is the propagation angle from the z-axis, and h is thepropagation angle from thex-axis. Let

a :kh cos/ cos h2pG cos/ cos h; b:kh cos/ sin h

2pG cos/ sin h; c:kh sin/

2pG sin/:

By a simple computation, we have the dispersion equation

k2h2L M; 3:1

in which

L :w1w23 H

w33 Fw4E;

M:2c13 H c2 HF c3 F 3E ;

with E:cos a cosb cos c; F :cos a cos b cosa cosc cosb cosc, and H:cos a cosb cosc:. Then, replacing k in the leftside of the Eq.(3.1)with the numerical wavenumber kN yields

kN

1

h

ffiffiffiffiffiM

L

r : 3:2

The next proposition presents the error between the numerical wavenumber kN and the exact wavenumberk.

Proposition 3.1. For the 27-point finite difference scheme(2.8)witha0, there holds

kN

k 1

24 c1 c2 8c3 b1

4 b24 b3

4h i

1

4 c3 c2 b1b2

2 b1b32 b2b3

2h i

3

8

c3 1

12

w21

6

w31

4

w4k3h2 Ok4h3; kh! 0; 3:3where b1 :cos/ cosh; b2 :cos/ sinh, and b3 :sin/.

Proof. Let s:kh. Then, ab1s; bb2s; cb3s; Es cos a cos b cos c; Fs cos a cosb cosa cosc cosb cosc, andHs cos a cosb cosc. Moreover, bothMandL depend on the variables, denoted byMsandLsrespectively. Withthe Taylor expansion, we have

Ms s2 1

12 c1 c2 8c3 b1

4 b24 b3

4h i

6 c2 c3 b1b22 b1b3

2 b2b32

h i 9c3

n os4 Os5;

3:4

and

1

Ls 1 1

6 w2 2w3 6w4 s2

Os3

: 3:5



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In addition, from the Eq.(3.2), we have

kNh

2MsLs

:

Together with Eqs.(3.4) and (3.5), we have

kN

2

k2

1

12 c1 c2 8c3 b1

4 b24 b3

4

h i1

2 c3 c2 b1b2

2 b1b32 b2b3

2

h i34c3

16w2

13w3

12w4

k4h2

Ok5h3

; kh! 0:

Based on the above equation, applying the Taylor expansion of the function ffiffiffiffiffiffiffiffiffiffiffiffi

1 sp at the point s0 yields the conclusion ofthis proposition. h

We remark that the above proposition indicates thatkN approximates kin a second order. Moreover, the term associatedwith k3h2 presents the pollution effect, which depends on the wavenumber k, the parameters of the finite difference formula(2.8)and the waves propagation angle defined by / and h.

4. A refined parameter choice strategy for the new 27-point finite difference scheme

In this section, a refined strategy is presented to choose optimal parameters for the new scheme (2.8), based on minimiz-

ing the numerical dispersion. An optimization problem is solved with a finer setting to estimate the weight coefficientsc1;c2;c3, andw1; w2; w3; w4.

As is known, normalized numerical phase velocity and group velocity are two important tools in measuring the numericaldispersion, and the former is usually preferred in practice (see[22,25,29,39]). For the Eq.(2.16)witha0, its normalizednumerical phase and group velocity are

VNphv

G

2p

ffiffiffiffiffiM

L

r ; 4:1

and

VNgrv

G

4pv

VNph

1h

@M@k

LM 1

h@L@k

L2

; 4:2

respectively. Withh2pGk, we can easily conclude that

kN

k

VNphv

G

2p

ffiffiffiffiffiM

L

r ; 4:3

which means minimizing the error between the numerical wavenumber kN and the wavenumber kis equivalent to minimiz-ing the normalized phase velocity.

Now, we choose optimal parameterscjj1;2;3andwjj1;2;3;4 by minimizing the numerical dispersion. To thisend, we set

Jc1;. . .;c3;w1;. . .;w4;G;/; h: G

2p

ffiffiffiffiffiM

L

r 1; 4:4

withP3

j1cj1;P4

j1wj1 andG;/; h 2IG I/ Ih. Here, IG; I/ and Ih are three intervals. In general, one can chooseI/ :

0; p2

; Ih :

0; p4

and IG :

Gmin; Gmax

4;400

with GminP 2 by the Nyquist sampling limit[29]. It is observed from

(4.3)that minimizing the error between the numerical wavenumber kN and the exact wavenumber k is equivalent to min-imizing the normkJc1; . . . ; c3; w1; . . . ; w4; ; ; k1;IGI/Ih . For this purpose, letting Jc1; . . . ;c3; w1; . . . ;w4; G;/; h 0 yields

c1 2G2 3 3EFH

h i c2 2G

2 3E 2FH h i

w1 4p2E 1

w2 4p2 E1

3H

w3 4p2 E

1

3F

4p2E 2G2 3EF : 4:5

We choose

1

G

1

Gl0:

1

Gmax l

0 1

1Gmin

1Gmax

l 1 2

1

Gmax; 1

Gmin

for l

01;2;. . .; l;

/ /

0

m :

m0 1p

2m 1 2I/ for m

0

1;2;. . .;m;



8/24

and

h hn0 :n0 1p4n 1

2Ih for n0 1; 2;. . .;n:

Letal0 ;m0 ;n0 : 2pGl0

cos/m0coshn0 ; bl0 ;m0 ;n0 : 2pGl0

cos/m0sinhn0 , andcl0 ;m0 : 2pGl0

sin/m0 . Then, from(4.5), we obtain a overdeterminedlinear system

S11;1;1 S21;1;1 S

31;1;1 S

41;1;1 S

51;1;1

..

. ... ..

. ... ..

.

S11;1;n S21;1;n S

31;1;n S

41;1;n S

51;1;n

S11;2;1 S21;2;1 S

31;2;1 S

41;2;1 S

51;2;1

..

. ... ..

. ... ..

.

S1l0 ;m0;n0 S2l0;m0 ;n0 S

3l0 ;m0;n0 S

4l0;m0 ;n0 S

5l0 ;m0;n0

..

. ... ..

. ... ..

.

S1l;m;n S2l;m;n S

3l;m;n S

4l;m;n S

5l;m;n

26666666666666666664

37777777777777777775

c1c2w1

w2

w3

26666664

37777775

S61;1;1

..

.

S61;1;n

S61;2;1

..

.

S6l0 ;m0;n0

..

.

S6l;m;n

26666666666666666664

37777777777777777775

; 4:6

where

S1l0 ;m0;n0 :2G

2l0 3 3El0 ;m0 ;n0 Fl0;m0;n0 Hl0;m0;n0

;

S2l0 ;m0;n0 :2G2l0 3El0;m0 ;n0 2Fl0;m0 ;n0 Hl0 ;m0;n0

;

S3l0m0;n0 :4p2El0 ;m0 ;n0 1;

S4l0 ;m0;n0 :4p2 El0;m0;n0

1

3Hl0 ;m0;n0

;

S5l0 ;m0;n0 :4p2 El0;m0;n0

1

3Fl0 ;m0;n0

;

S6l0 ;m0;n0 :4p2El0 ;m0;n0 2G

2l0 3El0;m0;n0 Fl0 ;m0 ;n0

;

El0;m0;n0 :cos al0 ;m0 ;n0cos bl0;m0 ;n0cos cl0;m0;

Fl0;m0 ;n0 :cos al0 ;m0;n0cos bl0;m0 ;n0 cosal0;m0;n0cos cl0;m0 cosbl0;m0;n0cos cl0 ;m0

Hl0 ;m0;n0 :cos al0;m0;n0 cosbl0;m0 ;n0 cos cl0 ;m0:The above coefficient matrix has l m nrows and 5 columns, and can be solved by the least-squares method. In [25],

optimal parameters were chosen globally, and were used in the computation for different frequencies, velocity and stepsizes. This may yield much numerical dispersion for large wavenumbers and varying kx;y;z. To reduce the numerical dis-persion and improve the accuracy of the difference scheme, we propose the following rule with a finer setting.

Rule 4.1 (Refined choice strategy). Step 1.Estimate the intervalIG : Gmin; Gmax .

Step 2.Choosecii2N3andwjj2N4such that

c1;. . .;c3;w1;. . .;w4 argmin kJc1;. . .;c3;w1;. . .;w4; ; k1;IGI/Ih :X3j1 cj 1;

X4j1wj 1

( ): 4:7

Dispersion reducing schemes have been discussed since the 1990s, and they can be found in the work of Tam and his col-laborators (cf.[3537]). Now, we shall present some relation between Tams work and ours. Both of the work are based onthe dispersion relation of the waves, that is, a functional relation between the angular frequency of the waves and the wave-numbers of the spatial variables. This relation is usually obtained by taking the space and time Fourier transforms of the gov-erning equations, and it determines all the dispersiveness, damping rate, isotropy or anisotropy, group and phase velocitiesof waves. Therefore, both the work of Tam and ours are based on the fact that a good finite difference scheme should have thesame or almost the same dispersion relation as the original equations. However, there are also some difference. Firstly, weconcentrate on solving the acoustic wave equation in the frequency domain, that is the Helmholtz equation, which possessessome different properties compared with the acoustic wave equation in the time domain. Tam and his collaborators focus onthe acoustic problems in the time domain, especially the linearized Euler equations. Secondly, we take a different dispersionminimizing strategy. For our finite difference scheme, we respectively compute the numerical dispersion Eq. (3.1)and thenormalized numerical phase velocity (4.1), which have seven weight parameters to be determined. As is known, the normal-ized numerical phase velocity would be 1, if there is no numerical dispersion. Then, we could obtain the weight parameters



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by minimizing the error between the normalized numerical phase velocity and 1. For the method of Tam and his collabora-tors, they try to formulate a finite difference scheme nearly having the same Fourier transform in space or time as the ori-ginal partial derivatives. The finite difference approximation of the spatial derivatives and the treatment for the timederivative were discussed separately. For details, we refer the reader to[37].

We now turn our attention to computing the weight parameters with Rule 4.7. In general, we can estimate IGby usingaprioriinformation before computing parameters. For constant wavenumbers, the number of grids per wavelength Gusuallylocates in 2;400, which can be partitioned into some small intervals for practical computing, such asIG :

2;2:5

;2:5;3

; . . .

10;400

. For varying wavenumbers, the velocity v locates in

vmin;vmax

, where vmin; vmax denote

the minimum and maximum velocity respectively. For each frequency f, we can obtain the interval IG Gmin; Gmax withGmin vminfh andGmax vmaxfh . Then, for each intervalIG, together withI/ : 0; p2andIh : 0; p4, we can obtain a group of param-eters. In computation, we need to solve a overdetermined linear system (4.6), whose size l m n 5 is related with thepartition ofIG; I/, and Ih. Generally, we would obtain better parameters with a finer partition, which means a more expensivecost. Fortunately, we do not need a very fine partition, since the benefit increases little with the increasing finer partition. Inpractice, we choose l20; m20; n10, and the resulting system can be solved not expensively.

Table 1 gives a group of refined parameters. In Fig. 2(a)(c), we present the normalized phase velocity curves for the stag-gered-grid scheme with global choice strategy (the staggered-grid 27p), the new scheme with global choice strategy (theglobal 27p), and the new scheme with refined parameters (the refined 27p) inTable 1, respectively. As can be seen, boththe global 27p and the refined 27p scheme have an improvement over the staggered-grid 27p scheme, and the refined27p scheme has the least numerical dispersion. Therefore, we would expect that the refined 27p scheme can be used to con-trol the numerical dispersion and to suppress nonphysical oscillations to a certain degree.

Numerically, the dispersion is not sensitive to the small change of the weight parameters. That is, the dispersion wouldnot have a big change if the parameters are disturbed a little. To see this, we present two dispersion pictures in Fig. 2(d) and(e), in which the parameters are taken from (c) but with perturbations of 0.1 and 0.01, respectively. As can be observed, for asmall change of parameters, the normalized phase velocity curve in (d) is almost the same as that of the original. However, ifparameters are disturbed too much, the normalized phase velocity curve would change significantly. We also point out that asmall perturbation of the parameters would not have great influences on the performance of the difference scheme. We shallpresent this by examples in Section7.1.

5. A 3D complex shifted-Laplacian-PML preconditioner for the 3D Helmholtz-PML equation

In this section, the complex shifted-Laplacian-PML preconditioner is developed for the 3D Helmholtz-PML equation, and aspectral analysis is given to the discrete preconditioned system from the perspective of linear fractal mapping on complexplane.

After discretization of the 3D Helmholtz-PML equation, we would like to solve the resulting linear system with iterativesolvers, since direct methods are limited by the storage. For the iterative method, Krylov subspace methods, such as the Bi-CGSTAB[41]and GMRES[27], are usually preferred. However, due to the indefiniteness and bad condition number of theresulting system, the Krylov subspace method is not competitive, and preconditioning is required. That is, a good precondi-tioner should be constructed to make the preconditioned system have a favorable spectral distribution, which contributes toa lower condition number and hence a fast convergence of the iterative method (cf.[7]).

Many preconditioned iterative solvers for the Helmholtz equation have been explored in the past few decades. In Baylisset al.[5],established a benchmark for the iterative solution of the Helmholtz equation. They proposed an iterative algorithmby using the preconditioner of Laplacian, combined with the conjugate gradient (CG) iteration. Since the Helmholtz problemwas indefinite and non-self-adjoint, the preconditioned CG iteration was applied to the normal equation which is symmetricand positive definite. The preconditioner was inverted approximately with one SSOR sweep. In [6,18], an even greaterimprovement was obtained when a multigrid sweep, plus a redblack ordering was employed to invert approximatelythe preconditioner. Later, a family of shifted-Laplacian preconditioners [14,15,23]were developed, which performed effi-ciently, especially the complex shifted-Laplacian preconditioner (cf.[14])

Table 1

Refined optimal parameters.

IG [2, 2.5] [3.5, 4] [5,6] [7, 8] [9, 10] [10,400]

c1 0.5035127 0.7617528 0.8159342 0.8354262 0.8432810 0.8269996c2 0.0720630 0.0148152 0.0340791 -0.0394517 0.0414069 4.097e07c3 0.4244243 0.2530624 0.2181449 0.2040255 0.1981258 0.1729999w1 0.4058413 0.7602512 1.1330134 1.7177071 2.4693294 2.9473150w2 0.1966284 0.4883334 1.5191327 3.2400262 5.4811311 6.8805122w3 0.5979158 1.1153920 2.1033335 3.8084643 6.0429826 7.4116566w4 0.2003855 0.3873097 0.7172142 1.2861453 2.0311809 2.4784594

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M: D b1b2ik2; 5:1

whereb1andb2are positive parameters. In[14,15,23], the multigrid sweep was also employed to invert the preconditionerapproximately. As can be seen, the shifted-Laplacian preconditioner is a generalization of the original Laplacian precondi-tioner proposed in[5]. It can be obtained by adding a zeroth order term to the Laplacian. For the preconditioning systemof shifted-Laplacian preconditioner, the preconditioned Krylov subspace method such as Bi-CGSTAB is employed, and is di-rectly applied to the discrete Helmholtz equation. However, for that of Laplacian preconditioner [5,6,18], they used the pre-

conditioned CG iteration, which was applied to the normal equation instead.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.85

0.9

0.95

1

1.05

1.1

Vph

N

/v

1/G

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.85

0.9

0.95

1

1.05

1.1

1/G

VN p

h/v

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.85

0.9

0.95

1

1.05

1.1

1/G

VN p

h/v

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.85

0.9

0.95

1

1.05

1.1

1/G

VN p

h/v

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.85

0.9

0.95

1

1.05

1.1

1/G

VN p

h/v

Fig. 2. Normalized phase velocity curves for: (a) the staggered-grid 27p scheme, (b) the global 27p scheme, (c) the refined 27p scheme, (d) the refined 27pscheme with parameters in (c) disturbed by 0.01. (e) the refined 27p scheme with parameters in (c) disturbed by 0.1.



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In this paper, for the 3D Helmholtz-PML Eq.(2.2), we base our preconditioner on the operator

M: @

@x A

@u

@x

@

@y B

@u

@y

@

@z C

@u

@z

b1b2iD: 5:2

Ifb1; b2 > 0, we call the operator(5.2)as the 3D complex shifted-Laplacian-PML preconditioner. When ABC1 andDk2,(5.2)is equivalent to the operator(5.1). The 3D complex shifted-Laplacian-PML preconditioner gives a nice clusteredspectrum, which will be presented later. We employ (5.2) to precondition the 3D Helmholtz-PML Eq. (2.2), obtaining the pre-

conditioned equation in the operator form

AM1vg; v Mu: 5:3

We now pay attention to the discretization of(5.3), and the spectral analysis of the discrete preconditioned system. Applyingthe new 27-point finite difference scheme(2.8)to discretize(5.3), we obtain the discrete preconditioned system

AM1

v g; v Mu; 5:4

whereA; M2CNN; u;g2 CN;C is the complex number set and Nis the number of unknowns. MatricesAand Mare the dis-crete Helmholtz-PML operator and preconditioner respectively, and they have the forms

A:Lz1D; z1 :1 ai; 5:5

and

M:Lz2D; z2 :b1 b2i; 5:6

whereL;Dare discretizations of the 3D Laplacian-PML @@x A @@x @@y B @@y @@zC @@z, and the operator corresponding to thezeroth order term, respectively. The coefficient matrix A is sparse and diagonal-distributional. Moreover, due to the PML,Ais nearly symmetric with complex entries. As is observed, preconditioner Mcan be constructed easily, just by discretizingM with the same difference scheme forA. The inverse ofM can be approximated by the multigrid method. The convergenceof the multigrid method is related to parameters b1 and b2 (especially b2), which also have an important influence on thespectral distribution of the preconditioned system(5.4). It follows from the spectral analysis later that the choice ofb2 isto strike a balance between the convergence rate of the multigrid method and a favorable spectrum of the preconditionedsystem(5.4).

We now study the spectrum of the discrete preconditioned 3D Helmholtz-PML Eq.(5.4). For the 2D Helmholtz equation,an analysis was given to the discrete preconditioned system in [17], with the Neumann, Dirichlet, and Sommerfeld boundarycondition. For the 2D Helmholtz-PML equation preconditioned with a 2D complex shifted-Laplacian-PML, we presented a

spectral analysis from the perspective of the linear fractional mapping on extended complex plane (see[12]). The spectralbehavior can be understood clearly in this manner. Hence, we continue our method proposed in [12] for the spectral analysisof(5.4). To do this, we denote the eigenvalues ofA byrAfor any matrixA2 CNN. We first recall a lemma in[12].

Lemma 5.1. LetL;D2CNN,A :Lz1DandM :Lz2Dwithz1;z22 C. IfD andM are nonsingular,l2 Candk : lz1lz2withl z2 then k2rAM1if and only ifl2rD1L, andM1A;D1Lshare the same eigenvectors.

With the help of above lemma, we can define a linear fractional mapping by

k kl:l z1l z2

: 5:7

Denote the real and imaginary part oflby Rel and Iml respectively. Then, relevant to(5.7), we have the following lem-ma, which is a generalization of Lemma 2.2 in [12].

Lemma 5.2. Let the linear fractional mappingk: C ! C be defined by (5.7). Then the linel lciwith c2R is mapped tothe circleOc; Rwith c:z1z2icz2z2icandR :

z1z2z2z2ic

, and the half-planesl l> ciandl l< ciare mapped inside andoutside Oc; R, respectively.

Proof. In the complex plane C fl: lx yi; x;y2Rg, a circle can be represented by

ax2 y2 bxcyd 0; 5:8

with parameters a; b; c;d2R satisfying aP 0 and b2 c2 >4ad. The center of this circle is b2a ; c2a and the radius isR:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2c24ad

p2a . Noting that

x

l l

2 ; y

l l

2i ; and x2

y2

ll;



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(5.8)can be rewritten as

all fl fl d 0; 5:9

with f: 12 b ci; jfj2 >ad. The center and the radius can be represented by c fa andRffiffiffiffiffiffiffiffiffiffiffijfj2ad

pa

respectively. Whena0,(5.9)is reduced to a line which can be considered as a circle with R 1on the complex plane. Now, we considerthe line

l lci; 5:10

which is parallel to the real axis. From(5.7), we have

lz2kz1k 1

; and lz2k z1

k 1 : 5:11

Substitution of(5.11)into(5.10)yields

a0kkf0k f0kd0

0; 5:12

wherea0 : z2 z2i c; f0 : z2z1i candd0 : z1 z1i c. Then,(5.12)represents a circle, denoted by Oc;Rwiththe center c: f0

a0z2z1ic

z2z2icand the radiusR :ffiffiffiffiffiffiffiffiffiffiffiffiffiffijf0 j2a0d0

pa0 z1z2z2z2ic

. It can be easily obtained thatl l> ciis mapped in-

sideOc; R, andl l< ciis mapped outside Oc; R. h

The following proposition immediately follows the linear fractional mapping(5.7)and the Lemma5.2.

Proposition 5.3. Letz1 :1 ai,z2 :1 bi(i.e., b11; b2bin (5.6)) with b> aP 0. Then the linel lciis mappedto the circle Oc; R with c: abc2bc andR: bac2bc, the half-plane l l> ci is mapped inside Oc;R, and the half-planel l< ciis mapped outside Oc; R. Moreover, for the discrete preconditioned system(5.4)and a constantr2 0;1, thereexists a positive constant b0such that whena< b 6 b0the spectrum forD

1Lis mapped inside the circle O1; r, that is, foranyl2rD1L; jkl 1j 6 r.

Proof. The first result of this proposition follows from the Lemma 5.2 withz1 :1 ai,z2 :1 bi, namely,b11; b2b in(5.6). The second result follows from lim

b!ak1. h

We next give an example to illustrate the above proposition explicitly. Fig. 3presents the linear fractional mapping(5.7)

with z1 :1a0andz2 :1 0:5ib0:5. As can be seen, lines l0 : l l0 (plotted in red), l1 : l l0:2i(green)andl2 : l l 0:2i(blue) are mapped into circles l00 : O12 ; 12(red),l

01 : O 914 ; 514(green) andl

02 : O38 ; 58(blue), respectively.

The linel0 is the real axis and its image l00 is tangent to the imaginary axis. The upper half-plane (the shadow region) and

the lower half-plane (white region) are mapped inside and outside of the circle l00 respectively. The points l1 1,l20:5; l31; l41:5 inFig. 3(a) are mapped to k11; k20:5 0:5i; k30; k40:5 0:5iinFig. 3(b), respectively.Moreover, all the three circles are tangent at k11.

From the above results, we can see thatrAM1 would be enclosed by a certain circle in the right-half plane if and only ifrD1L has a nonnegative imaginary part. In [12], for 2D case, a series of numerical experiments were made to prove numer-

Fig. 3. The linear fractional mapping.



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ically that there holds the sufficient condition, namely, the spectrum ofD1Lhas a nonnegative imaginary part. However, for

the 3D case, we can not obtain that all of the eigenvalues for D1Lhave a nonnegative imaginary part. Hence, not all of the

eigenvalues for AM1 would be enclosed by the red circle. Fig. 4 shows the spectrum for A; D1L and AM1 withk

15;z1

1;z2

1

0:5iand h

124. Fig. 4(a) presents the original spectrum for A, which is scattered over both the left

and right half-planes. As can be seen, the real part of the spectrum includes a part of the negative real axis with large values,which means A is seriously indefinite. After preconditioning, for AM1, a nice clustered spectrum is observed in Fig. 4(c),which is far away from the origin with all of the eigenvalues in the right half-plane. It is seen that most of the spectrum

forAM1 is enclosed by the circle O12 ; 12 (plotted in red), with few eigenvalues lying outside of the circle. We can expect thisfrom the spectrum ofD1LinFig. 4(b), which shows that most of the eigenvalues are located on the upper half-plane.Fig. 5shows the corresponding spectrum withh;k;z2the same asFig. 4, butz11 0:05i, namely, with 5% damping in(2.2). Thespectrum inFig. 5(b) is more clustered due to a nonzero damping a.

As is known, the Krylov subspace method is particularly efficient for the system with an Hermitian positive definite ma-trix, or more generally, for system with all eigenvalues of the coefficient matrix in the right half of the complex plane. In

Fig. 4,though some eigenvalues are not enclosed by the circle O12 ; 12, the spectrum ofAM1 is still favorable enough forthe convergence of Krylov subspace methods. Moreover, we can properly choose the parameter b to make the spectrum

ofAM1 closer to one. In fact, for certain A;Mwith z1 :

1ai and z2 :

1

bi, according to Proposition5.3, there exists

someb0> 0, such that whena < b < b0; jk 1j is smaller than a certain positive number. Hence, the spectrum ofAM1 inFig. 4 contributes to a fast convergence of the Krylov space method, which is validatedby numerical experiments in Section 7.Nevertheless, it should be pointed out that b should be chosen properly in order to achieve a faster convergence of the Krylov

subspace method. When b is too small, though a good spectrum ofAM1 can be obtained, it is difficult to invert approxi-mately the preconditioner Mwith the multigrid method, due to its poor property. On the other hand, since lim

b!1k0, when

bis too large, we would get an unfavorable spectrum near the origin, which leads to a poor performance of preconditioning.

Fig. 6shows the spectrum ofAM1 forb2; b0:8 and b0:2, respectively. Both the cases inFig. 6would weaken theefficiency of the Krylov subspace method for solving the preconditioned system (5.4). Hence, the choice ofbis a compromisebetween the convergence rate of the multigrid method and a favorable spectrum of the preconditioned system(5.4). In thistext, we choose b0:5 in(5.2), which is a proper choice in practice.

Fig. 4. The spectral distribution for matrices (a) A, (b) D1L, (c) AM1.

Fig. 5. The spectral distribution for A and AM1 withh1=24; k15; z11 0:05iandz21 0:5i: (a) A; (b) AM1.

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6. A 3D full-coarsening multigrid method in the preconditioned Bi-CGSTAB

In this section, we develop a 3D full-coarsening multigrid method to invert the preconditioner M approximately, whileusing the preconditioned Bi-CGSTAB to solve the discrete preconditioned system. For the full-coarsening strategy, we pro-

pose a new matrix-based prolongation operator.For solving the Helmholtz problem, the Bi-CGSTAB method is usually preferred (cf. [14]). As can be seen in Section5, thespectrum of the discrete preconditioned system(5.4)is favorable for the convergence of Bi-CGSTAB method. In the precon-ditioned Bi-CGSTAB algorithm, we can not afford to invert exactly the preconditioner M in the computation ofu :M1v,where vis the given vector with u to be determined. Hence, we choose to solve the additional linear system Muv. In orderto obtain a better approximate solution at a lower cost, the multigrid method is employed to solve the additional linear sys-tem. Thus, the preconditioned Bi-CGSTAB algorithm combines Bi-CGSTAB and the multigrid method, which are considered asthe outer iteration and inner iteration respectively. Now, we consider solving the additional linear system with the multigridmethod. For the multigrid method, the traditional linear prolongation operator serves well for solving Muvwith constantwavenumbers. However, for varying wavenumbers, it is not robust enough, and leads to a divergence of the multigrid-basedpreconditioned Bi-CGSTAB method. Here, we consider a matrix-based prolongation operator, which is constructed based onthe finite difference stencil, according to the algebraic information of the coefficient matrix. In[44], a matrix-based prolon-gation operator was developed to handle the 2D convectiondiffusion problem. In[14], the multigrid based on Zeeuws pro-

longation operator was used to invert approximately the preconditioner for the 2D Helmholtz problem. Later, theprolongation operator was extended to the 3D case with some modification in[26,42], where the 3D multigrid was basedon the semi-coarsening strategy, performing coarsening only in two directions while keeping the third direction unchanged.

In this paper, we intend to employ the multigrid method based on a 3D full-coarsening strategy, plus a pointwise smooth-er which is easy to implement. The use of a pointwise smoother is to avoid the alternating plane relaxation which is veryexpensive, because a sub-system has to be solved in every plane. For the full-coarsening prolongation operator, we have

Fig. 7. One coarse and eight fine grid cells with capital letters indicating coarse grid points and small letters indicating fine grid points.

Fig. 6. The spectral distribution ofAM1 with (a) b2; (b) b0:8; (c) b0:2.

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to take into account the prolongation in the third direction, which need not be considered in the semi-coarsening case. Moti-vated by Zeeuws 2D prolongation operator, we propose here a new 3D full-coarsening matrix-based prolongation operator,which gives a good performance.

To describe the construction of the new prolongation operator, we begin withFig. 7, which shows one coarse and eightfine grid cells with capital letters indicating coarse grid points and small letters indicating fine grid points. The set of finegrids denoted by Xh is split into eight disjunct subsets, that is,

Xh fXh;0;0;0;Xh;1;0;0;Xh;0;1;0;Xh;1;1;0;Xh;0;0;1;Xh;1;0;1;Xh;0;1;1;Xh;1;1;1g:

Here, Xh;0;0;0 consists of fine grid points which are also coarse grid points. Xh;1;0;0;Xh;0;1;0 and Xh;0;0;1 consist of fine gridpoints which are located between two coarse grid points along the x-, y- and z-axis, respectively. Xh;1;1;0;Xh;1;0;1 andXh;0;1;1 consist of fine grid points which are located respectively in x-y;x-zand y-zplane but not next to any coarse gridpoints. Xh;1;1;1 consists of fine grid points which are not next to any coarse grid points. For example, in Fig. 7,A;2 Xh;0;0;0, p2Xh;0;1;0; q2Xh;1;0;0; r2 Xh;1;1;0; a02 Xh;0;0;1;p02 Xh;0;1;1; q02 Xh;1;0;1 and r02 Xh;1;1;1. Symbols eh andeHrepresent the grid functions for the fine and coarse grid respectively. Assuming eHis already known on coarse grids, the en-tries ofeh on fine grids can be obtained by prolongation among the nearest coarse grid neighbors. For example, the prolon-gation weights at p are denoted by WAp; WCp, and the component ofeh on p is obtained by prolongation from A andCaccording toWAp;WCp. Then, the 3D full-coarsening prolongation operator can be expressed by

ehA:e2hA;

ehp:WApe2hA WCpe2hC;

ehq:WAqe2hA WBqe2hB;

ehr:WAre2hA WBre2hB WCre2hC WDre2hD;

eha0:WAa

0e2hA WA0 a0e2hA

0;

ehp0 :WAp

0e2hA WCp0e2hC WA0 p

0e2hA0 WC0 p

0e2hC0;

ehq0:WAq

0e2hA WBq0e2hB WA0 q

0e2hA0 WB0 q

0e2hB0;

ehr0:WAr

0e2hA WBr0e2hB WCr

0e2hC WDr0e2hD WA0 r

0e2hA0

WB0 r0e2hB

0 WC0 r0e2hC

0 WD0 r0e2hD

0;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

6:1

where A2Xh;0;0;0;p2Xh;0;1;0; q2Xh;1;0;0, r2 Xh;1;1;0; a02 Xh;0;0;1;p02 Xh;0;1;1;q02 Xh;1;0;1; r02 Xh;1;1;1, andWA; WB; WC; WD;WA0 ; WB0 ; WC0 ; WD0 denote the weights to be determined on each fine grid. We can also re-write(6.1)in the form of matrixvector multiplication, that is ehPeH, whereP is the prolongation matrix (prolongationoperator) to be determined.

Now, we describe our new 3D full-coarsening matrix-based prolongation operator, which is constructed based on the 27-

point stencil, that is, the algebraic information of the coefficient matrix. In one coarse and eight fine grid cells, prolongationweights should be derived from three kinds of fine gridpoints, denoted by K1 Xh;0;1;0

SXh;1;0;0

SXh;0;0;1;K2

Xh;1;1;0SXh;1;0;1

SXh;0;1;1 andK3 Xh;1;1;1, respectively. Denote the 27-point difference stencil for preconditionerMby

M,

M1;1;1 M0;1;1 M1;1;1

M1;0;1 M0;0;1 M1;0;1

M1;1;1 M0;1;1 M1;1;1

264

375z1

M1;1;0 M0;1;0 M1;1;0

M1;0;0 M0;0;0 M1;0;0

M1;1;0 M0;1;0 M1;1;0

264

375z

M1;1;1 M0;1;1 M1;1;1M1;0;1 M0;0;1 M1;0;1

M1;1;1 M0;1;1 M1;1;1

264 375z1

: 6:2

We first describe the determination of the prolongation weights on fine gridpoints in K1. Taking the point a 0 inFig. 7forexample, we have

eha0:WAa

0e2hA WA0 a0e2hA

0;

withWAa0; WA0 a0to be determined. AssumingMto be the difference stencil ata0, thenM0;0;0is the entry ofMon the cen-tral pointa0. IfMl;m;1or Ml;m;1(l;m2 f1;0;1g; l;m 0;0) is not zero, it means a coupling between the value ofux;y;zat a0and the value at the gridpoint corresponding to Ml;m;1or Ml;m;1. These couplings should be incorporated in the construc-tion of prolongation weights WAa0 andWA0 a0. Experiments show that it would weaken the efficiency of the multigridmethod when neglecting either Ml;m;1or Ml;m;1. Without splitting Minto symmetric and anti-symmetric parts, we give a sim-

ple formula



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r1 :Xl;m

Ml;m;1

; d1 :maxfjMl;m;1jg; l;m2 f1;0;1g:

r2 :Xl;m

Ml;m;1

; d2 :maxfjMl;m;1jg; l;m2 f1; 0; 1g:

j:maxfr1; d1g; i:maxfr2; d2g;

and

WAa0

jj i

; WA0 a0

ij i

: 6:3

We can similarly compute the prolongation weights on other points in K1. The difference stencil Mhere is asymmetricdue to the PML and varying wavenumber k in heterogenous medium, and we do not split M(orM) into symmetric andanti-symmetric parts, as is done in the construction of prolongation operator of Zeeuw-type [14,26,42]. The prolongationoperator of Zeeuw-type was originally developed in[44]to handle the convectiondiffusion problem. The coefficient matrixwas split into symmetric and anti-symmetric parts which were considered to originate from the diffusion and convectionterms respectively. In this text, the multigrid method is used to invert approximately the preconditioner M for the Helmholtzproblem, and it is not absolutely necessary to splitMinto symmetric and anti-symmetric part as done for handling the con-vectiondiffusion problem.

After obtaining weights on gridpoints in K1, we now describe the determination of prolongation weights on fine grid-points in K2and K3. Using the multigrid to solve Mu

v, after several relaxation sweeps, an approximate solution denoted

by ~uis obtained, and the residual is ~r: v M~u. After the coarse grid correction, we get a correction solution u: ~u eh~u PeH, and its residual error is

r: v Mu ~r MPeH: 6:4

As in the 2D case of diffusion problems [1], in order to prevent huge jumps in the norm of the residual after prolongation, werequire

MPeHjK2 0: 6:5

Then, for the fine gridpoint in K2 such as r, we have

X1l1

X1m1

X1n1

Ml;m;nreh;l;m;nr 0: 6:6

Here, ifl; m; n 0;0;0; eh;l;m;nr denotes the component ofeh restricted on r, otherwise it denotes the component re-stricted on the neighbor ofr. We rewrite(6.6)as

X1l1

X1m1

Ml;m;0reh;l;m;0r X1l1

X1m1

Xn1;1

Ml;m;1reh;l;m;nr 0: 6:7

Assume that we have obtained the prolongation

eh;1;0;0r:WBte2hB WDte2hD;

eh;0;1;0r:WCse2hC WDse2hD;

eh;1;0;0r:WApe2hA WCpe2hC;

eh;0;1;0r:WAqe2hA WBqe2hB:

8>>>>>: 6:8

Substituting (6.8)andeh;0;0;0

r

into the first part of(6.7)and neglecting the second part, then we can determine weights

WAr;WBr; WCrandWDrineh;0;0;0r:WAre2hA WBre2hB WCre2hC WDre2hD:

However, the second part of(6.7)should not be neglected in order to get a good prolongation at r. To this end, we leteh;l;m;1r eh;l;m;nr eh;l;m;0r, and substitute them into the second part of(6.7), yielding

X1l1

X1m1

X1n1

Ml;m;neh;l;m;0r 0; 6:9

which is an approximation of(6.7). From(6.9), we can compute weights WAr; WBr; WCrandWDr, which would im-prove the performance of the multigrid. In the meantime, (6.9)also contributes to a 2D lumped stencil along the z-axis.Denoting



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Mzl;m :X1n1

Ml;m;n; l;m2 f1;0; 1g;

we have the 2D lumped stencil along the z-axis as

Mz,

Mz1;1 Mz0;1 M

z1;1

Mz1;0 Mn0;0 M

z1;0

Mz

1;1 Mn

0;1 Mz

1;1

2

664

3

775; 6:10

and the 2D lumped stencil along n-axis (nx;y) as

Mn ,

Mn1;1 Mn

0;1 Mn

1;1

Mn1;0 Mn0;0 M

n1;0

Mn1;1 Mn0;1 M

n1;1

2664

3775; 6:11

whereMxm;n :P1

l1Ml;m;n, Myl;n :

P1m1Ml;m;n; l; m; n2 f1;0;1g.Fig. 8shows the 2D lumped stencil along the x-axis. In

the semi-coarsening case of[26], the lumped stencil in one direction is similarly used. Thus, we complete the constructionof prolongation weights for K1

SK2. Substituting all the weights already obtained into(6.5)yields prolongation weights on

r02 K3, without splitting(6.6)into two parts and approximating it. Up to now, we have finished the construction of the pro-longation operator.

We remark that the construction of the 3D full-coarsening prolongation operator is based on the difference stencilMand2D lumped stencils along thex;y;z-axis respectively. That is, we use the algebraic information of the coefficient matrix M. Ascan be seen, prolongation weights are derived similarly in three directions, and they are used to handle the variation ofwavenumberkx;y;zalongx;y;z-axis respectively. In order to prevent huge jumps of the norm of the residual after prolon-gation, 2D lumped stencils in three directions are used, which shall contribute to a more robust and efficient prolongationoperator in practice. For the 3D Laplacian, with classical 7-point difference stencil on fine grids without PML, 2D lumpedstencils reduce to the classical 5-point stencil, and prolongation weights for p; q; a0;p0; q0; r and r0 reduce to 12 ;

12 ;

12 ;

14 ;

14 ;

14,

and18, respectively, which is just the trilinear interpolation. It is well known that the trilinear prolongation operator leadsto an efficient and robust multigrid in this classical setting.

7. Numerical experiments

In this section, the numerical convergence of the new difference scheme is tested, and some comparisons are made with

the compact sixth order difference scheme proposed in[34]. Then, numerical experiments related to geophysical applica-tions are presented to show the efficiency and robustness of the multigrid-based preconditioned Bi-CGSTAB method forthe discrete preconditioned system(5.4). The multigrid is based on the 3D full-coarsening strategy with the new prolonga-tion operator. We employ the full multigrid V-cycle (FMG), which possesses both the robustness of the W-cycle and the effi-ciency of the V-cycle [9]. One FMG iteration with two relaxation sweeps is enough. For multigrid components, the pointwise

Jacobi relaxation with underrelaxation (x-JAC) is used as a smoother, which is easy to parallel. We adopt the full weighting

z

xy

m0,0,1

x

m0,1,1

x

m0,1,1

x

m0,0,0

x

m0,0,1

x

m0,1,0

x

m0,1,1

x

m0,1,1

x

m0,1,0

x

Fig. 8. The construction of the lumped stencil along x-axis.



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restriction operator, the newly proposed prolongation operator, and the coarse grid operator obtained by the Galerkin prin-ciple. For the restriction operator, instead of using the transposition of the prolongation operator, we choose the weightingoperator suggested in[14],which gives a better performance here. The experiments range from constant wavenumber tovarying wavenumber in heterogenous medium. In the computation, a zero initial guess has been used, and the precondi-tioned Bi-CGSTAB iteration terminates when the Euclidean norm of the relative residual error is reduced to the order of106. All the experiments here are performed serially on an Intel Xeon (8-core) with 3.33 GHz and 96 Gb RAM. Moreover,Matlab 7v is also used as the testing platform.

7.1. Numerical convergence and comparison of the new scheme

In this section, we shall present the numerical convergence of the the new scheme, whose effect is also shown with aperturbation of the weight parameter. Then, we compare the scheme with the sixth order compact in [34], which was ob-tained by compact reformulation of the Helmholtz equation. For convenience, we consider the following 3D Helmholtz equa-tion with zero Dirichlet boundary condition, which was similarly used in[34],

Duk2u f; in0;13; 7:1

where f : 3n2 1 sinnkx sinnky sinnkz, with kmp; m;n1;2;3; . . .. Then, the exact solution isux;y;z sinnkx sinnky sinnkz=k2. Tables 2 and 3 give numerical convergence of the new scheme with its variationcaused by the perturbation of parameters for k5p; m1 and k12p; m4 respectively. C.O. denotes the numericalconvergence order with error in the normj j1. OP, PPS and PPL represent the new scheme using the original parameters,and parameters obtained by a perturbation of the original by 0:01 and 0:1, respectively. As can be seen, the new schemeis of second order convergence, and a small perturbation of the parameter has a small influence on the scheme, while a largeperturbation significantly influences its convergence.

In Table 4, the new scheme is compared with the sixth order compact scheme in [34]. We can observe that when the steph1=20 is fixed, for small wavenumbers, the sixth scheme outperforms the new second order scheme, but for large wave-numbers, the new second order scheme gives a better performance. In fact, the sixth scheme would definitely outperformthe new second order scheme, so long as the step is small enough. However, in practice, the stephis related with the wave-number k, especially for large wavenumbers. Due to the storage limit, the number of gridpoints per wavelength Gcan not betoo large, that is, step h can not be too small. In addition, the pollution analysis of error shows that the accuracy not onlydepends on the convergence order, but also the wavenumber. In this setting, the new second order scheme may performbetter, since it can reduce the numerical dispersion. InTable 5, fork30pandn10, we present the numerical error cor-responding to both schemes with different steph. We can see that the new second order scheme performs better for a largewavenumber, ifh remains not too small. We point out that both the sixth and the new second order scheme use 27 grid-

points, but they are developed based on different motivation. After second order approximation, there are still lots of degreesof freedom left. With these degrees, the sixth scheme aims to construct a formula of high accuracy, assuming the wavenum-ber is constant, the steps in all direction are equal, and the solution and source term are smooth enough. However, the newsecond order scheme combines these degrees with the relation of dispersion to minimize the numerical dispersion. So, thenew second order scheme is optimal in the sense of suppressing the dispersion, while the sixth scheme is optimal in thesense of convergence order.

We now turn our attention to the preconditioned BI-CGSTAB method for solving the linear systemAug, which is ob-tained by discretization of the Helmholtz Eq.(7.1). The preconditionerMis based on the 3D complex shifted-Laplacian. Weshall examine the effect of three different discretizations of the Helmholtz and preconditioning operator. The first scheme,denoted by SC62, uses the 6th difference scheme in[34]to discretize the Helmholtz operator, and uses the common 7-pointdifference scheme to discretize the preconditioning operator. Two other schemes, denoted by SN62 and NN22, respectivelyuse the 6th scheme in [34]and the new 2nd scheme in this paper to discretize the Helmholtz operator, while both using thenew 2nd scheme to discretize the preconditioning operator. We do not employ the 6th scheme to discretize the precondi-

tioning operator, because the obtaining of the 6th scheme involves a right-hand term. Fig. 9 shows spectra ofAM1

, based onSC62, SN62 and NN22, respectively, with k4p; n5pandh1=16. As can be seen, SC62 gives a poor spectrum ofAM1,since some of the eigenvalues locates in the left half-plane, which is unfavorable for the convergence of the BI-CGSTAB meth-od. Both SN62 and NN22 contribute to a good spectral distribution ofAM1, which are clustered in the right half-plane. In

Table 2

The numerical convergence of the new scheme with its variation caused by perturbation of the parameters, for k 5p; n 1.

h 1/4 1/8 1/16 1/32 1/64

OP Error 1.50e2 3.70e3 8.59e4 2.22e4 5.58e5C.O. 2.02 2.10 1.96 1.99

PPS Error 1.51e2 3.80e3 1.00e3 3.89e4 9.92e5C.O. 1.99 1.93 1.92 1.97

PPL Error 3.34e

2 1.46e

2 6.10e

3 2.80e

3 1.44e

3

C.O. 1.19 1.25 1.14 0.96



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computation, since the discretization of the preconditioning operator for SN62 is based on the 7-point scheme, it indeedsaves multiplication operations. However, the effect of preconditioning is not good, that is, it costs too many preconditionedBI-CGSTAB iterations, which coincides with the spectrum inFig. 9(a). For SN62 and NN22, since both discretizations of the

Helmholtz and preconditioning operator are based on 27-point scheme, they cost the same multiplication operations for

Table 6

Number of Bi-CGSTAB iterations and CPU time in minutes (in parentheses) for different wavenumbers k with 2% and without damping.

Grid k= 60 k= 100 k= 140 k= 180 k= 220

973 1613 2243 2883 3523

Ia0:00 43 (1.73) 60 (7.38) 83 (23.30) 108 (58.68) 152 (140.19)a0:02 32 (1.38) 42 (5.27) 55 (15.65) 70 (38.36) 89 (83.74)IIa0:00 45 (1.92) 64 (8.68) 90 (27.84) 119 (70.36) 168 (169.06)a0:02 34 (1.57) 46 (6.13) 61 (18.98) 78 (46.27) 99 (100.53)

0.5 0 0.5 1 1.51

0.8

0.6

0.4

0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 10.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 10.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 9. The spectra ofAM1 based on (a) SC62, (b) SN62, (c) NN22.

Table 3

The numerical convergence of the new scheme with its variation caused by perturbation of the parameters, for k 12p; n 4.

h 1/8 1/16 1/32 1/65 1/129

OP Error 1.49e1 3.61e2 8.60e3 2.20e3 5.46e4C.O. 2.02 2.10 1.96 1.99

PPS Error 1.35e1 3.75e2 9.70e3 2.50e3 6.38e4C.O. 1.85 1.95 1.98 1.97

PPL Error 7.39e2 4.10e2 1.14e2 8.00e3 4.32e3C.O. 0.85 1.84 0.51 0.88

Table 5

Comparison of the error for the new and sixth schemes when k 30p and n 10.

h 1=20 1=40 1=80 1=160

6th Error 16808 1081 21.76 0.4128C.O. 3.96 5.63 5. 72

2nd Error 0.0710 0.0442 0.0115 0.0029C.O. 0.6838 1.94 1.98

Table 4

Comparison of the new scheme with the sixth order scheme, for different k with n 5 and h 1=20.

k 2p 3p 4p 5p 6p 7p 8p

G2pkh

22.0 14.7 11.0 8.8 7.3 6.3 5.5

Error for 6th 0.0014 0.0097 0.0389 0.1258 0.3468 1.315 20.69Error for 2nd 0.0116 0.0126 0.0206 0.0250 0.0286 0.0559 0.4969



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each preconditioned BI-CGSTAB iteration, and the total iterations make little difference. In Section 7.2to Section7.4, weadopt the scheme NN22, that is, both the Helmholtz operator A and the preconditioning operatorM are discretized bythe new second order scheme.

7.2. The 3D Helmholtz-PML equation with constant wavenumber

We consider a domain X 0;13, with point and line sources placed at different locations of the domain. The number ofgridpoints per wavelength is chosen to be G= 10, and an accuracy requirement for second order discretizations is thatkh 6 2pG p5. The thickness of the PML is set to be 20, that is, the PML possesses 20 gridpoints in each direction. Table 6pre-sents the number of preconditioned Bi-CGSTAB iterations and the CPU time needed (in parentheses) in minutes, with andwithout damping for different wavenumbers respectively. The largest wavenumber for test is k220, and the number ofunknowns (without PML) is 3523 43;614;208 (about 43.6 millions).

In Tables 6, (I) represents the preconditioned BI-CGSTAB method based on the 3D full-coarsening multigrid with the new-ly proposed prolongation operator, plus a point-wise x-JAC smoother. (II) represents the preconditioned BI-CGSTAB meth-ods based on the 3D semi-coarsening multigrid with the Zeeuws prolongation operator, plus a line-wise x-JAC smoother.Fork220 without damping, it needs 152 iterations for method (I), and the CPU time needed is about 140 min. As can beseen from the table, the method (I) gains a faster convergence than (II). Moreover, the CPU time needed for each iteration of(I) is less than that of (II). This is justified by the fact that the multigrid with semi-coarsening method has larger coarse grid

operators, compared with the counterparts in full-coarsening method. It can also be observed that the convergence speedwith some damping in the Helmholtz problem is faster than that without damping, and we can expect this from the spectraldistribution in Fig. 5 (b), which is more clustered. Fig. 10 presents the number of iterations for the preconditioned Bi-CGSTABmethod versus wavenumber k. We keep khp5, which means 10 gridpoints per wavelength is used. In Fig. 10, a nearly linearincrease is observed in the number of iterations with the increase of wavenumber k, especially for the damping case. Thenumerical solution corresponding to k60 without damping (a0) is presented inFig. 11, with point and line sourcesplaced at different locations of the domain.

7.3. The three layers model

The three layer model is used to evaluate the preconditioned Bi-CGSTAB for a simple heterogeneous medium, in whichcase the wavenumberkis slightly varying due to the varying velocityvin different medium. We consider a physical domainwhich is defined to be a cube of dimension 760 m

760 m

760 m. A point source is located at the upper surface with

(60 m, 60 m, 0 m), where the upper surface is assigned to bez0.Fig. 12presents the domain with three layers, and thevariation of velocity in the medium. InFig. 12(a), the first, second, and third layer represent 1600 m/s, 2400 m/s, 3200 m/srespectively. The real part of the numerical solution atf30 Hz without damping is plotted inFig. 12(b).Table 7presentsthe convergence of the preconditioned Bi-CGSTAB method for different frequencies (varying from 20 to 60 Hz) with 2% andwithout damping. The number of Bi-CGSTAB iterations and CPU time in minutes are presented. As can be seen, method (I)performs robustly and still achieves a faster convergence than method (II).

7.4. The 3D salt dome model

We evaluate a more complicated heterogeneous medium, namely, the 3D salt dome model, which mimics the subsurfacegeology under the sea. The physical domain considered here is defined to be a cube of dimension 960 m 960 m 600 m. Apoint source is located at point (481 m,481 m,257 m) under the upper surface, which is assigned to bez0.Fig. 13(a) pre-sents the 3D salt dome model, and the variation of velocity in different medium. Fig. 13(b) and (c) present the cross-sectionof the 3D salt dome model at x481 m and y549 m respectively. The velocity of sound is irregularly structured

60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

Wavenumber k

Numberofiterations

Method (I), =0.00

Method (I), =0.02

Method (II), =0.00

Method (II), =0.02

Fig. 10. The number of iterations for preconditioned Bi-CGSTAB versus wavenumber k.

http://-/?-http://-/?-http://-/?-http://-/?-


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throughout the domain. The lowest velocity is 1524 m/s and the highest velocity is 4480 m/s, which is in the salt dome (thered part in Fig. 13). The snapshots of the real part of the numerical solution forf 30 Hz without damping atx481 m andy549 m are displayed inFig. 14. We can observe the wave filed propagating from the source (481 m, 257 m) through themodel.Table 8 presents the preconditioned Bi-CGSTAB convergence for the case with and without damping for different fre-

quencies, which vary from 20 Hz to 40 Hz. The numbers of Bi-CGSTAB iterations and CPU time in minutes (in parentheses)

Fig. 11. The real part of numerical solution fork60 with point and line sources placed at different locations. (a) point source at the center of the domain;(b) 2D cross-section of (a) along thex-axis; (c) point source on the edge (z-axis); (d) point source on the corner; (e), (f), (g) point sources on the upper faceand right face; (h), (i) line sources in the directions ofx-axis andy-axis.

1.6km/s

2.4km/s

3.2km/s

x

z

y

Fig. 12. (a) The three layers model with speed profile indicated. (b) The real part of numerical solution at f 30 Hz.



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x (m)

z(m)

120 240 360 480 600 720 840 960

120

240

360

480

600

y (m)

z(m)

120 240 360 480 600 720 840 960

120

240

360

480

600

Fig. 13. (a) The 3D salt dome model with velocity profile indicated; (b) and (c) the cross-section of 3D salt dome model at x481 m andy549m,respectively.

x (m)

z(m)

120 240 360 480 600 720 840 960

120

240

360

480

600

y (m)

z(m)

120 240 360 480 600 720 840 960

120

240

360

480

600

Fig. 14. Monofrequency wavefield (real part) for f 30 Hz without damping at (a) x481 m ; (b)y549 m, respectively.

Table 7

The number of Bi-CGSTAB iterations and CPU time in minutes (in parentheses) for the three layers model with 2% and without damping.

Grid f= 20 f= 30 f= 40 f= 50 f= 60

893 1213 1853 2323 2963

(I)a0:00 48 (1.58) 61 (3.78) 79 (13.66) 105 (32.63) 141 (81.96)a0:02 43 (1.50) 52 (3.34) 65 (11.47) 82 (25.68) 110 (65.25)(II)a0:00 50 (1.79) 66 (4.61) 90 (16.66) 118 (39.64) 159 (100.67)a0:02 45 (1.68) 56 (3.88) 74 (13.59) 91 (31.56) 122 (78.32)



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are shown. Forf 40 Hz, the interior domain contains 257 257 161 10;633;889 gridpoints (about 10.6 million). Withthe PML, the unknowns amount to 297 297 201 17;730;009 (about 17.7 million) in total. The numbers of iterationsfor method (I) and (II) without damping are 158 and 180, with the time-consuming being about 42 and 53 min, respectively.For the complicated heterogeneous medium, method (I) still achieves a robust performance, and outperforms method (II).Even though both method (I) and (II) have the same iterations, method (I) has an advantage over method (II). Becausethe semi-coarsening strategy in method (II) leads to a gradual decrease of the coarse grid operators, and more time wouldcost in each iteration of the preconditioned Bi-CGSTAB method.

8. Conclusion

In this paper, we study the numerical solver for the 3D Helmholtz equation with a boundary of PML. Both the finite dif-ference scheme and preconditioned iterative solver are concerned.

For the discretization of 3D Helmholtz-PML equation, we have developed a new 27-point finite difference scheme, whichis second order. The new scheme has been proved to be consistent with the 3D Helmholtz-PML equation, and equivalent tostaggered-grid 27-point scheme under certain conditions. The classical dispersion analysis has been made to obtain theapproximation of numerical wavenumber to the exact wavenumber. Based on minimizing the numerical dispersion, a re-fined choice strategy is taken to optimize parameters of the new difference scheme. Comparisons of phase velocity showthe improvement of the new scheme with refined optimal parameters.

After discretization of the 3D Helmholtz-PML equation with the new difference scheme, we obtain a sparse indefinite lin-ear system. To solve the linear system, the preconditioned Krylov subspace method Bi-CGSTAB has been employed, and thepreconditioner is based on the 3D complex shifted-Laplacian-PML preconditioner, which is generalized from the 2D shifted-

Laplacian preconditioner. A spectral analysis is given from the perspective of fractional linear mapping in complex variablefunction. In order to invert approximately the preconditioner, the multigrid method has been used, which is based on the 3Dfull-coarsening strategy, plus a pointwise Jacobi smoother. A new matrix-based prolongation operator has also beenconstructed for the 3D full-coarsening multigrid. Numerical experiments have been presented, ranging from constantwavenumbers to highly varying wavenumbers in heterogeneous medium. Numerical results illustrate the efficiency ofthe full-coarsening multigrid-based preconditioned Bi-CGSTAB method.

References

[1] R. Alcouffe, A. Brandt, J. Dendy, J. Painter, The multigrid method for the diffusion equation with strongly discontinuous coeffcients, SIAM J. Sci. Statist.Comput. 2 (1981) 430454.

[2] I. Babuka, F. Ihlenburg, E. Paik, S. Sauter, A genralized finite element method for solving the Helmholtz equation in two dimensions with minimalpollution, Comput. Methods Appl. Mech. Engrg. 128 (1995) 325359.

[3] I. Babuka, S. Sauter, Is the pollution effect of the FEMavoidable for the Helmholtz equation considering high wave numbers, SIAM Rev. 42 (2000) 451

484.[4] G. Baruch, G. Fibich, S. Tsynkov, E. Turkel, Fourth order schemes for time-harmonic wave equations with discontinuous coefficients, Commun. Comput.Phys. 5 (2008) 442455.

[5] A. Bayliss, C. Goldstein, E. Turkel, An iterative method for Helmholtz equation, J. Comput. Phys. 49 (1983) 631644.[6] A. Bayliss, C. Goldstein, E. Turkel, The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics, Comput.

Math. Appl. 11 (1985) 655665.[7] M. Benzi, Preconditioning techniques for large linear systems: a survey, J. Comput. Phys. 182 (2002) 418477.[8] J. Brenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185200.[9] W. Briggs, V. Henson, S. McCormick, A Multigrid Tutorial, Second ed., SIAM, California, 2000.

[10] Z. Chen, H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J.Numer. Anal. 41 (2003) 799826.

[11] Z. Chen, T. Wu, H. Yang, An optimal 25-point finite difference scheme for the Helmholtz equation with PML, J. Comput. Appl. Math. 236 (2011) 12401258.

[12] Z. Chen, D. Cheng, W. Feng, T. Wu, H. Yang, A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML, J. Math.Anal. Appl. 383 (2011) 522540.

[13] J. Douglas Jr., D. Sheen, J. Santos, Approximation of scalar waves in the space-frequency domain, Math. Mod. Methods Appl. Sci. 4 (1994) 509531.[14] Y. Erlangga, C. Oosterlee, C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput. 27 (2006) 1471

1492.[15] Y. Erlangga, C. Vuik, C. Oosterlee, On a class of preconditioners for the Helmholtz equation, Appl. Numer. Math. 50 (2004) 629651.

Table 8

The number of Bi-CGSTAB iterations and CPU time in minutes (in parentheses) for the 3D salt dome model with 2% and without damping.

Grid f= 20 f= 25 f= 30 f= 35 f= 40

1292 81 1612 101 1932 121 2252 141 2572 161(I)a0:00 57 (3.01) 72 (6.17) 89 (11.95) 120 (23.64) 158 (42.51)a0:02 55 (2.95) 67 (5.84) 81 (10.90) 106 (21.03) 139 (37.92)(II)a0:00 61 (3.52) 79 (6.68) 102 (14.95) 137 (29.27) 180 (53.46)a0:02 58 (3.41) 73 (6.17) 91 (13.42) 118 (25.83) 157 (47.89)



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[16] X. Feng, H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal. 47 (2009) 28722896.[17] M. Van Gijzen, Y. Erlangga, C. Vuik, Spectral analysis of the discrete Helmholtz operator with a shifted Laplacian, SIAM J. Sci. Comput. 29 (2007) 1942

1958.[18] J. Gozani, A. Nachshon, E. Turkel, Conjuate gradient coupled with multigrid for an indefinite problems, in: R. Vichnevestsky, R.S. Tepelman (Eds.),

Advances in computer methods for partial differential equation, vol. V, IMACS, New Brunswick, NJ, 1984, pp. 425-427.[19] B. Hustedt, S. Operto, J. Virieux, Mixed-grid and staggered-grid finite difference methods for frequency-domain acoustic wave modelling, Geophys. J.

Int. 157 (2004) 12691296.[20] F. Ihlenburg, I. Babuka, Finite element solution of the Helmholtz equation with high wave number, Part I: The h-version of the FEM, Comput. Math.

Appl. 30 (1995) 937.[21] F. Ihlenburg, I. Babuka, Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer.

Methods Eng. 38 (1995) 42074235.[22] C. Jo, C. Shin, J. Suh, An optimal 9-point, finite difference, frequency-space, 2-D scalar wave extrapolator, Geophysics 61 (1996) 529537.[23] A. Laird, M. Giles, Preconditioned iterative solution of the 2D Helmholtz equation, Report 02/12, Oxford Computer Laboratory, Oxford, UK, 2002.[24] Y. Luo, G. Schuster, Parsimonious staggered grid finite differencing of the wave equation, Geophys. Res. Lett. 17 (1990) 155158.[25] S. Operto, J. Virieux, P. Amestoy, J. LExcellent, L. Giraud, H. Ali, 3D finite difference frequency-domain modeling of visco-acoustic wave propagation

using a massively parallel direct solver: a feasibility study, Geophysics 72 (2007) 195211.[26] C. Riyanti, A. Kononov, Y. Erlangga, C. Vuik, C. Oosterlee, R. Plessix, W. Mulder, A parallel multigrid-based preconditioner for the 3D heterogeneous

high-frequency Helmholtz equation, J. Comput. Phys. 224 (2007) 431448.[27] Y. Saad, W. Schultz, GMRES: a generalized minimal residual algorithmfor solving nonsymmetric linear system, SIAM J. Sci. Statist. Comput. 7 (1986)1

176.[28] E. Saenger, N. Gold, A. Shaoiro, Modeling the propogation of elastic wave using a modified finite difference grid, Wave Mot. 31 (2000) 7792.[29] C. Shin, H. Sohn, A frequency-space 2-D scalar wave extrapolator using extended 25-point finite difference operator, Geophysics 63 (1998) 289296.[30] I. Singer, E. Turkel, A perfectly matched layer for the Helmholtz equation in a semi-

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A Dispersion Minimizing Finite Difference Scheme and Precond - Chen Et Al - 2012