An Explicit Method Based on the Implicit Runge–Kutta ...

Bulletin of the Seismological Society of America, Vol. 99, No. 6, pp. 3340–3354, December 2009, doi: 10.1785/0120080346

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations by Dinghui Yang, Nian Wang, Shan Chen, and Guojie Song

Abstract A new explicit differentiator series method based on the implicit Runge– Kutta method, called the IRK-DSM in brief, is developed for solving wave equations. To develop the new algorithm, we first transform the wave equation, usually described as a partial differential equation (PDE), into a system of first-order ordinary differential equations (ODEs) with respect to time t. Then we use a truncated differentiator series method of the implicit Runge–Kutta method to solve the semidiscrete ordinary differential equations, while the high-order spatial derivatives included in the ODEs are approximated by the local interpolation method. We analyze the theoretical properties of the IRK-DSM, including the stability criteria for solving the 1D and 2D acousticwave equations, numerical dispersion, discretizing error, and computational efficiency when using the IRK-DSM to model acoustic-wave fields. For comparison, we also present the stability criteria and numerical dispersion of the so-called Lax–Wendroff correction (LWC) methods with the fourth-order and eighth-order accuracies for the 1D case. Promising numerical results show that the IRK-DSM provides a useful tool for large-scale practical problems because it can effectively suppress numerical dispersions and source-noises caused by discretizing the acoustic- and elastic-wave equations when too-coarse grids are used or the models have a large velocity contrast between adjacent layers. Theoretical analysis and numerical modeling also demonstrate that the IRK-DSM, through combining both the implicit Runge–Kutta scheme with good stability condition and the approximate differentiator series method, is a robust wave-field modeling method. Introduction Numerical methods of solving wave equations have proven to be successful and provided useful tools in exploration seismology. Many numerical methods of synthetic seismograms have been developed and successfully applied to solve practical geophysical problems. These methods include finite-difference (FD) methods, such as high-order compact difference or so-called Lax–Wendroff correction (LWC) schemes (e.g., Kelly et al., 1976; Dablain, 1986; Virieux, 1986; Levander, 1988; Fornberg, 1990; Lele, 1992; Igel et al., 1995; Blanch and Robertsson, 1997; Geller and Takeuchi, 1998; Takeuchi and Geller, 2000; and many others); the finite-element method (Yang et al., 2008); the adaptive finite-element method (Turner et al., 1956; Whiteman, 1975; Johnson, 1990; Eriksson and Johnson, 1991); the reflectivity method (Fuchs and Müller, 1971; Booth and Crampin, 1983a, b); the pseudospectral method (PSM) (Kosloff and Baysal, 1982; Kosloff et al., 1984; Vlastos et al., 2003); and the spectral element method (Priolo and Seriani, 1991; Priolo et al., 1994; Komatitsch and Vilotte, 1998; Komatitsch et al., 2000), and so on. Each method has its advantages and disadvantages. For example, the

high-order finite-difference method is widely used because of its good properties, such as high accuracy and easy implementation for computer code, but it usually suffers from serious numerical dispersions when a coarse grid is used or the models have a large velocity contrast between adjacent layers (Wang et al., 2002). The finite-element method can deal with irregular region and curve boundary, whereas it needs to solve large-scale linear algebraic equations, resulting in both the large storage-space requirement and the costly central processing unit (CPU) time. The reflectivity method can only be applied to media with homogeneous horizontal layers. The space operators of the pseudospectral method fit the Nyquist frequency, but it needs Fourier transform that is time-consuming and the usage of a global basis, resulting in inaccuracies in wave fields for models with strong heterogeneity or sharp boundaries (Komatitsch et al., 2000). Numerical dispersion is an important issue in numerical seismic simulations and has been widely studied by many researchers (Alford et al., 1974; Sei and Symes, 1994; Fei and Larner, 1995; Zhang et al., 1999; Yang et al., 2002).

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An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations Roughly speaking, numerical dispersion is an unphysical phenomenon caused by discretizing the wave equation (Sei and Symes, 1994; Yang et al., 2002). Such an unphysical phenomenon affects our recognition of seismic-wave propagation. A natural idea in dealing with numerical dispersion is to use higher accurate methods with local difference operator (Yang et al., 2006). However, the numerical dispersion or undesirable ripple affects also the performance of the socalled high-order FD methods. For example, the tenth-order compact FD schemes (e.g., Dablain, 1986; Wang et al., 2002) that usually use more grids than low-order schemes also suffer from numerical dispersions. The demand of more grids in high-order FD methods reduces the efficiency of the algorithm for parallel implementation and prevents it from artificial boundary treatment such as the PSM. The fluxcorrected transport (FCT) technique can lessen or eliminate the numerical dispersion, but it is unable to fully recover lost resolution caused by suppressing the numerical dispersion when a too-coarse grid is used or the geological models have a large velocity contrast between adjacent layers (Fei and Larner, 1995; Yang et al., 2002; Zheng et al., 2006). The so-called nearly analytic discrete method (NADM) and its improved versions were developed by Yang et al. (2003, 2007) for reducing the numerical dispersion caused by the discretization of acoustic- and elastic-wave equations. These methods use the wave displacement, the velocity, and their gradient fields simultaneously to restructure the wave displacement-fields. Hence, they can effectively suppress the numerical dispersion. However, the NADM and its improved version have a relatively small Courant number defined by α
c0 Δt=Δx with the acoustic velocity c0 (Dablain, 1986; Sei and Symes, 1994), which limits the stability range of numerical calculations, whereas the optimal NADM (ONADM) (Yang et al., 2006) cannot be applied to the two-phase porous wave equations including the so-called particle velocity ∂U=∂t of the wave displacement U because the ONADM does not compute the velocity fields. The stability criterion α
c0 Δt=Δx ≤ αmax (αmax denotes the maximal Courant number) gives the relationship between temporal step, spatial increment, and wave velocity to keep the numerical calculation stable, and it provides the theoretical guide for numerical calculations. It is well-known that implicit methods, such as alternative direction implicit method for solving PDE or diagonal implicit Runge–Kutta method (IRK) for solving ODE, have promising stability criteria in theory. However, we usually need to solve large-scale linear algebraic equations when we use the implicit methods to solve wave equations, resulting in large storage requirements and increasing the CPU time. The explicit method has the limitation of stability conditions, but it is easy to be implemented; its computational speed is fast at each time-advancing step. An explicit algorithm based on an implicit method may have advantages of both explicit and implicit methods. The main purpose of this article is to propose an explicit Runge–Kutta differentiator series method (IRK-DSM), based on both the implicit Runge–Kutta method for solving the

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ODE and the differentiator series method that was initially proposed to solve Cauchy problems or the linear ODE or PDE (Ke, 1996; Ke and Xie, 1999). To do this, we first trans-

form the wave equations into a system of semidiscrete ODEs, and then change the diagonal implicit Runge–Kutta method for solving the semidiscrete ODEs into an explicit algorithm (IRK-DSM) via the truncated differentiator series. On the basis of such a structure, the IRK-DSM has good computational stability and can effectively suppress the numerical dispersion and source-noises caused by discretizing the wave equations when too-coarse grids are used. This allows us to use larger time and spatial increments in the IRK-DSM for modeling seismic-wave fields, resulting in reducing computational CPU time and saving the storage for computer code.

The Explicit IRK-DSM Based on the Implicit Runge–Kutta Method Transform of Wave Equations In a heterogeneous elastic medium, the wave equation can be written as: ρ

∂ 2U
∇ · σ  f σ
C∶ε ∂t2 1 ε
∇U  ∇UT ; 2

(1)

where ρ denotes the density; U
u1 ; u2 ; u3 T , the displacement vector; σ and ε, the second-order symmetric stress and strain tensors, respectively; C, the fourth-order stiffness tensor; and f
f1 ; f2 ; f3 T , the external source force. We can rewrite equation (1) as follows: ρ

∂ 2U
D · U  f; ∂t2

(2)

where D is the second-order partial differential operator. For example, in the 2D anisotropic case D is defined by D




 ∂ ∂ ∂ ∂ ∂ ∂ C1  C2  C3  C4 ; ∂x ∂x ∂z ∂z ∂x ∂z

(3)

where 2

c11

6 C1
4 c16

c16 c66

c15

3

2

c15

7 6 c56 5; C2
4 c56

c14 c46

c13

3

7 c36 5;

c15 c56 c55 c55 c45 c35 2 3 2 3 c15 c56 c55 c55 c45 c35 6 7 6 7 C3
4 c14 c46 c45 5; C4
4 c45 c44 c34 5; c13

c36

c35

c35

and cij x; z are the elastic constants.

c34

c33

(4)

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D. Yang, N. Wang, S. Chen, and G. Song

∂u1 ∂u2 ∂u3 T Let W
∂U ∂t
 ∂t ; ∂t ; ∂t  ; then equation (2) can be written as:

∂U
W ∂t

∂W 1 1
D · U  f: ∂t ρ ρ

(5)

Let V
U; WT ; we can further rewrite equation (5) as the following ∂V
L · V  F; ∂t where

 L


0 I 1 ρD 0



(6)


and F


 0 : 1 ρf


V
ni;j  1  2rΔtK ni;j  rΔtK
ni;j  K
ni;j
L  Fi;j tn  1  rΔt; (11) p where r
12  63. Obviously, equations (10) and (11) imply that we need to solve two systems of linear algebraic equations at each step of time advancing when we use the implicit algorithm based on equations (9)–(11) to compute V
n1 i;j , resulting in an increment of the computational cost. To avoid solving the system of linear equations in the implicit method, we propose to use an explicit scheme in the IRK method. We rewrite equation (10) as follows
ni;j
L
· V
ni;j  Fi;j tn  rΔt: I  rΔtLK

(12)


< 1, then using the Taylor expansion, we have If krΔtLk From equation (6), we have the following equations: ∂ 2V ∂t∂x


L · Vx

∂2V ∂t∂z


1 L
· V
ni;j  Fi;j tn  rΔt K ni;j
I  rΔtL
L · Vz;

(7)

(8)

Owing to the local elastic property of rocks, we use the local interpolation method (Kondoh et al., 1994; Yang et al., 2003, 2007) to approximate the high-order derivatives ∂ kl U=∂xk ∂zl ni;j and ∂ kl W=∂xk ∂zl ni;j 2 ≤ k  l ≤ 3, including in the right-hand side of equation (8) through using the values of the wave displacement, the particle velocity, and their gradients at the grid point (i, j) and its neighboring grid points. These computational formulae of approximating the second- and third-order derivatives are listed in Appendix A. After discretizing the high-order spatial derivatives at the right-hand side of equation (8), equation (8) becomes a semidiscrete ODE. Now let us take a closer look at the semidiscrete ODE (8). In this article, we use the diagonal implicit Runge–Kutta (IRK) method (Hairer et al., 1993) to solve the ODE (8) as follows


V
ni;j  rΔtK ni;j   Fi;j tn  rΔt; K ni;j
L


m L
· V
ni;j  Fi;j tn  rΔt: rΔtL

(13)

Using the truncated Taylor series to approximate the righthand side of equation (13), we can further obtain the following equation K ni;j


2 X


m L
· V
ni;j  Fi;j tn  rΔt: rΔtL

(14)

m
0

Obviously, calculating K ni;j in equation (14) is an explicit computational formula. Similarly, we can obtain the following approximation, deriving from equation (11),

Formulation of the Explicit IRK-DSM

Δt n n1 V
i;j K i;j  K
ni;j ;
V
ni;j  2

∞ X m
0

where V x
∂V=∂x and V z
∂V=∂z. Define V

V; V x ; V z T , F

F; 0; 0T , and L

diagL; L; L. From equations (6) and (7), we derive the following equation: ∂ V


L
· V
 F: ∂t




(9)

(10)

K
ni;j


2 X


m fL
V
ni;j  1  2rΔtK ni;j  rΔtL

m
0

 Fi;j tn  1  rΔtg:

(15)

Combining equation (9) with equations (14) and (15), we obtain the explicit IRK-DSM. In this IRK-DSM, equations (14) and (15) involve the third-order operator L
3. It means that we need to compute the high-order derivatives k
∂zl ni;j k  l ≥ 4 as we use formulae (14) and ∂ kl V=∂x (15) to compute K ni;j and K
ni;j . To avoid computing these high-order spatial derivatives, we use a split-operator method to compute L
3 · V
ni;j and L
3 · V
ni;j  1  2rΔtK ni;j  . To do this, we rewrite equations (14) and (15) as follows n n K ni;j
V

i;j  rΔtL
2 · V
ni;j  rΔt2 L
2 · V

i;j



2 X m
0


m · Fi;j tn  rΔt; rΔtL

(16)

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations K
ni;j


2 X

Error Analysis


m fL
· V
ni;j  1  2rΔtK ni;j  rΔtL

m
0

Theoretical Analysis

 Fi;j tn  1  rΔtg


2 X

ml


m fL
· W
ni;j  Fi;j tn  1  rΔtg rΔtL

m
0



n  rΔtL
2 · W
n  rΔt2 L
2 · W

n
W i;j i;j i;j 

2 X


m · Fi;j tn  1  rΔtg; frΔtL

(17)

m
0 n

n are defined by where V

i;j , W
ni;j , and W i;j n V

i;j
L
· V
ni;j ;

(18)

W
ni;j
V
ni;j  1  2rΔtK ni;j ;

(19)



n
L
· W
n ; W i;j i;j

(20)

and the second-order operator L
2 can be obtained from the definition of L
as follows L
2
DiagL2 ; L2 ; L2    1 1 1 1 1 1
Diag D; D; D; D; D; D : ρ ρ ρ ρ ρ ρ

3343

(21)

For practical calculations, the implementation of the explicit IRK-DSM is divided into three major steps. The computational steps are described next. 1. Computing K ni;j by equation (16). n (a) Use formulae (A3) to (A9) to compute V

i;j via equation (18); n (b) Use the obtained result V

i;j in the first step (a) and apply the similar computational formulae as (A3) n to (A9) to compute L
2 · V

i;j and L
2 · V
ni;j via equation (21); (c) Substitute these results obtained in steps (a) and (b) into equation (16) to obtain K ni;j . 2. Computing K
ni;j by equation (17). In this step, we first compute W
ni;j using equation (19), and then use the similar steps as computing K ni;j in the first step to compute n K
ni;j . The difference is that these vectors V
ni;j and V

i;j presented in equation (16) are replaced respectively by the

n when computing K
n . vectors W
ni;j and W i;j i;j 3. Substitute these obtained results K ni;j and K
ni;j into equan1 at the (n  1)th tion (9) to obtain the values of V
i;j time level.

Note that the fourth terms at the right-hand side of equations (16) and (17) can be easily computed because of the known analytical source function F.

∂ V Using the Taylor series expansion, the error of ∂x m ∂zl 2 ≤ m  l ≤ 3 is OΔx4  Δz4  caused by the interpolation formulae presented in Appendix A. Because of the usage of the third-order implicit Runge–Kutta method and the truncated differentiator series method for solving the ODE (8), the temporal error, caused by the discretization of the temporal partial derivative, is OΔt3 . Therefore, we conclude that the error introduced by the IRK-DSM is OΔt3  Δx4  Δz4 . In other words, the IRK-DSM developed in this article has a fourth-order accuracy in space and third-order accuracy in time.


Numerical Error To further illustrate the accuracy of our present method, in the following discussion we will compare the numerical results of the IRK-DSM against other methods, such as the conventional FD and the fourth-order LWC methods, with the exact solution of the 2D acoustic-wave equation for the homogeneous medium. Under this consideration, we choose the following 2D initial problem: ∂2u ∂2u 1 ∂2u  2
2 2; 2 ∂x ∂z c ∂t

(22a)


 2πf0 2πf0 x cos θ0  z sin θ0 ; (22b) u0; x; z
cos  c c
 ∂u0; x; z 2πf0 2πf0 x cos θ0  z sin θ0 ;
2πf0 sin  c c ∂t

(22c) where c is the velocity of the plane wave; θ0 , the incident angle at time t
0; f0 , the frequency. The exact solution of this initial problem is:
  x z ut; x; z
cos 2πf0 t  cos θ0  sin θ0 : (23) α α In this numerical experiment, we choose the number of grid points N
201; the frequency, f0
15 Hz; the velocity, c
4000 m=sec; and θ0
π=4. The relative error for the 2D case is defined by  1 Er %
PN PN 2 ut n ; xi ; zj  i
1 j
1 1 N X N X 2 n 2 × ui;j  utn ; xi ; zj  × 100; (24) i
1 j
1

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D. Yang, N. Wang, S. Chen, and G. Song

where uni;j is the numerical solution and utn ; xi ; zj  is the analytical solution. Figures 1, 2, and 3 plot the relative errors Er versus time for different spatial and temporal increments, where three lines of Er corresponding to the IRK-DSM, the fourth-order LWC method, and the second-order FDM are shown in a semilog scale. In these figures, the maximum relative errors for different cases are listed in Table 1. From these error curves and Table 1 (Δx
Δz
h), we find that Er increases corresponding to the increase in the temporal and/or spatial increments for all the three methods. Figures 1, 2, and 3 illustrate that the IRK-DSM achieves the highest numerical accuracy among all three methods.

Stability Criteria It is well-known that the temporal increment Δt must be less than or equal to the Courant limit to keep the numerical calculation stable. In this section, following the ideas by Richtmyer and Morton (1967) and Guan and Lu (2006), we derive the stability criteria of the IRK-DSM for 1D and 2D cases. Through a series of mathematical operations, we obtain the following stability condition for the 1D homogeneous case (see Appendix B) c

Δt ≤ αmax ≈ 0:968 Δx

(25)

or Δt ≤ 0:968

Δx : c

(26)

Figure 1. The relative errors of the IRK-DSM, the fourth-order LWC method, and the second-order FDM measured by Er (formula [24]) are shown in a semilog scale for the 2D initial problem (22). The spatial and the temporal increments are 20 m and 2 × 104 sec, respectively.

Figure 2.

The relative errors of the IRK-DSM, the fourth-order LWC method, and the second-order FDM measured by Er (formula [24]) are shown in a semilog scale for the 2D initial problem (22). The spatial and the temporal increments are 30 m and 4 × 104 sec, respectively.

where c denotes the wave velocity, and Δt and Δx, the time and space increments, respectively. The maximal value αmax Δt of the Courant number defined by α
c Δx (Dablain, 1986; Sei and Symes, 1994) can be found in Appendix B. For comparison, we also give the stability conditions of the fourth-order and eighth-order LWC methods (Dablain, 1986) in Table 2 for the 1D case. From Table 2 we can see

Figure 3.

The relative errors of the IRK-DSM, the fourth-order LWC method, and the second-order FDM measured by Er (formula [24]) are shown in a semilog scale for the 2D initial problem (22). The spatial and the temporal increments are 40 m and 1 × 103 sec, respectively.

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

3345

Table 1 Comparison of Maximum Er % for Different Cases and Different Methods Methods

Case 1: Case 2: Case 3:

h
20 m Δt
2 × 104 sec h
30 m Δt
4 × 104 sec h
40 m Δt
1 × 103 sec

Second-Order FDM

63.2395 142.881 209.775

the stability condition of the IRK-DSM is more relaxed than those of the fourth-order and eighth-order LWC methods. For the 2D homogeneous case, the stability condition of the IRK-DSM under the condition Δx
Δz
h is given by (see Appendix B) Δt ≤ αmax

Δx Δx ≈ 0:844 : c c

(27)

These stability criteria for a heterogeneous medium cannot be directly determined but could be approximated by using a local homogeneous method. Our conjecture is that equations (26) and (27) are approximately correct for a heterogeneous medium if the maximal value of the wave velocity c is used.

Dispersion Analysis and Efficiency Almost all numerical modeling methods suffer from the numerical dispersion when too-coarse grids or too few samples per wavelength are used. We may use finer grid to increase the number of grid points per wavelength or increase the accuracy of numerical method to suppress or eliminate the numerical dispersion or artifacts. Using fine grid results in more computational costs and storages (the total number of grid points for a fixed-size model will rapidly increase, especially in 3D cases). It is difficult to apply these techniques in large-scale computation, especially for large-scale 3D simulation of seismic-wave propagation because of its intensive use of central processing unit (CPU) time and its need for large amounts of direct-access memory. Higherorder FD or compact FD (e.g., eighth-order method, Dablain, 1986), and staggered-grid FD (Virieux, 1986; Fornberg, 1990) can reduce the numerical dispersion, but merely increasing the accuracy of methods does not result in a proportional decrease of the numerical dispersion (Yang et al., 2006). In other words, the higher-order FD and staggeredgrid FD methods with local operators still suffer from the numerical dispersion when too few samples per wavelength Table 2 Approximate Maximum Courant Numbers of Different Methods for the 1D Case Methods

IRK-DSM

OΔt4  Δx4  LWC

OΔt4  Δx8  LWC

αmax

0.968

0.95

0.929

Fourth-Order LWC

IRK-DSM

0.899355

0.246253

4.83214

1.40869

14.4097

4.55115

are used (Sei and Symes, 1994; Yang et al., 2002). Therefore, the numerical dispersion is one of the most important issues which have to be solved in seismic simulations. In this section, following the analysis methods presented in the cited references (Dablain, 1986; Yang et al., 2006), we investigate the numerical dispersion caused by the IRK-DSM through the numerical dispersion relation for the 1D case. In the following discussion, we show the numerical results of the IRK-DSM using the dispersion relation (C3) that is found in Appendix C. For comparison, we also show numerically the dispersion relations of the fourth-order and eighth-order LWC methods. The detailed numerical dispersion analysis of the LWC methods can be found in Yang et al. (2006). Figures 4, 5, and 6 plot the dispersion relations of the IRK-DSM (formula C3) and the fourth-order LWC and eighth-order LWC methods (formula C5 presented in BSSA, Yang et al., 2006) for the 1D case, which show the ratio R of the numerical velocity to the real phase velocity versus θ
kΔx (where k is the wavenumber) for different methods. In other words, Figures 4–6 show the dispersion errors of the IRK-DSM, the fourth-order LWC, and eighth-order LWC, respectively. From Figure 4 we can see that the numerical velocity of the IRK-DSM gradually approximates the exact wave velocity when the Courant number α increases in the frequency range. It also shows that at the stability limit Δt (α
c Δx ≈ 0:968), the numerical dispersion of the IRKDSM is minimal. This suggests that we can minimize the numerical dispersion caused by the IRK-DSM by using the maximum Courant number. Moreover, the change of numerical dispersion corresponding to the change of the Courant number α is not sensitive as compared with the fourth-order and eighth-order LWC methods. This property is useful in practical calculations because we do not need to consider how to choose an appropriate Courant number in the stability range. From Figures 4 and 5 we observe that the numerical velocity for the IRK-DSM and the fourth-order LWC method are usually smaller than the exact velocity; this shows that the numerical dispersion of the IRK-DSM and the fourthorder LWC method follow the exact signal. However, as we observe in Figure 6, the numerical dispersion of the eighthorder LWC method is irregular for a fixed Courant number. It sometimes leads the exact signal and sometimes follows the signal. This phenomenon indicates that it is hard to choose a suitable Courant number for the eighth-order LWC.

3346

D. Yang, N. Wang, S. Chen, and G. Song

Figure 4.

The ratio R of the numerical velocity to the phase velocity versus wavenumber θ
kΔx for the IRK-DSM, where four lines correspond to α
0:3, 0.6, 0.8, and αmax , respectively.

Comparing Figure 4 with Figures 5 and 6, we can see that for the same Courant number, the numerical velocity of the IRKDSM approximates the exact wave velocity best, especially in the high-frequency range. It illustrates that the IRK-DMS has the least numerical dispersion among the three methods. We now investigate the numerical dispersion and computational efficiency of the IRK-DSM through modeling seismic-wave fields for the 2D case. For this case, we consider the following acoustic-wave equation in a 2D homogeneous medium:
2  ∂ 2u ∂ 2u 2 ∂ u
c   f; 0 ∂t2 ∂x2 ∂z2

Figure 5.

(28)

The ratio R of the numerical velocity to the phase velocity versus wavenumber θ
kΔx for the fourth-order LWC method, where four lines correspond to α
0:3, 0.6, 0.8, and αmax , respectively.

Figure 6.

The ratio R of the numerical velocity to the phase velocity versus wavenumber θ
kΔx for the eighth-order LWC method, where four lines correspond to α
0:3, 0.6, 0.8, and αmax , respectively.

where the source, located at the center of the computation domain, is an explosive source and the time variation of the source function is f
sin2πf0  exp4π2 f20 t2 =16 (Zahradnik et al., 1993), and has a Ricker wavelet with the peak frequency of f0
25 Hz. The acoustic velocity is c0
4000 m= sec, and the computational domain is 0 < x ≤ 24 km and 0 < z ≤ 24 km. Figure 7 shows the wave-field snapshots at t
2 sec on a coarse grid (Δx
Δz
80 m), generated respectively by the LWC methods with accuracies of orders 4 and 8, and the IRK-DSM with the same Courant number α
0:65 and the 0
2. We can see same grid points per wavelength G
f0c·Δx that the wave fronts of seismic waves shown in Figure 7, simulated by the three methods at the same time, are nearly identical; it took the fourth-order LWC, the eighth-order LWC, and the IRK-DSM about 6 sec, 8 sec, and 73 sec to generate Figures 7a,b,c, respectively. This means the computational cost of the IRK-DSM is more expensive than that of the high-order LWC methods with the same time steps and grid points for a fixed-size model. In other words, under the condition of the same grid interval and time step, our IRK-DSM method is more expensive than high-order LWC methods because our method computes more quantities (not only the wave field itself, but also its time derivative, the spatial gradients, etc.). However, Figure 7c, generated by the IRK-DSM, shows that the IRK-DSM has much less numerical dispersion even when the spatial increment is chosen by Δx
Δz
80 m, whereas the fourth-order LWC and eighth-order LWC methods exhibit serious numerical dispersions (see Figs. 7a,b). To reduce or eliminate the serious numerical dispersions generated by the high-order LWC methods, here we set the Courant number α
0:65 constant and diminish the space increment of the high-order LWC methods, then the grid

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

3347

Figure 8. Snapshots of seismic-wave fields at time 2 sec on a fine grid, generated by (a) the fourth-order LWC method (Δx
Δz
10 m), and (b) the eighth-order LWC method (Δx
Δz
15 m).

Figure 7. Snapshots of seismic-wave fields at time 2 sec on the coarse grid (Δx
Δz
80 m), generated by (a) the fourth-order LWC method, (b) the eighth-order LWC method, and (c) the IRKDSM, respectively. points per wavelength of the fourth-order LWC and the eighth-order LWC methods increased. Figure 8 shows the wave-field snapshots at t
2 sec on a fine grid and under the same Courant number of α
0:65, generated by the fourth-order LWC method (Δx
Δz
10 m, G
c0 f·Δx
20), and the eighth-order LWC method (Δx
Δz
15 m, G
16). Comparison between Figure 7c and Figure 8 demonstrates that the IRK-DSM with large grid increments (Δx
Δz
80 m, G
2) can effectively suppress the numerical dispersion, whereas high-order LWC methods have to use smaller grid increments to suppress numerical dispersion. The computation time for the IRK-DSM when larger grids are used is much less than the times by the highorder LWC methods when smaller grids are used for suppressing the numerical dispersion. Speaking in detail, in this experiment example it took the IRK-DSM about 1.22 min to generate Figure 7c, whereas the high-order LWC methods took about 62.77 min and 20.55 min to generate Figures 8a,b, respectively. It means that the computational speed of the IRK-DSM when the big grid is used is roughly 51 times that of the fourth-order LWC and about 17 times that of the eighth-order LWC method on a fine grid to achieve the same dispersion accuracy of the IRK-DSM. Meanwhile, the storage space required for computation in the IRK-DSM when the big grid is used is also different from that of the high-order LWC methods when smaller grids are used for suppressing the numerical dispersion. The IRK-DSM needs n1 n1 , wi;j , gradients of 24 arrays to store uni;j , wni;j , ui;j the displacement and the particle velocity, and the middle variables of K ni;j and L
· V
ni;j at each grid point, and the

number of grid points is 301 × 301 on a coarse grid for generating Figure 7c. Both of the high-order LWC methods n1 , uni;j , and only need 3 arrays to store the displacement ui;j n1 ui;j at each grid point, but the number of grid points on a fine grid for generating Figures 8a and 8b goes up to 2401 × 2401 for the fourth-order LWC and 1601 × 1601 for the eighth-order LWC, respectively. It indicates that the space storage of the IRK-DSM requires only about 12.6% of space storage of the fourth-order LWC method and roughly 28.2% of that of the eighth-order LWC method.

Numerical Modelling In this section we investigate the efficiency of the IRK-DSM for the 2D isotropic case and the transversely isotropic model with a vertical symmetry axis (VTI). It

should be mentioned in our present numerical experiments that wave velocities vary with propagation directions in anisotropic media, so the stability condition Δt ≤ 0:844h=vmax obtained previously is chosen where h (if Δx
Δz
h) corresponds to the spatial increment and vmax is the maximum P-wave or qP-wave velocity in the following models. All our experiments are implemented on a 2-core pentium 4 computer with 1G memory.

Isotropic Model Under this case of our consideration, the elastic-wave equation is ∂ 2 u1 ∂ 2 u1 ∂ 2 u1 ∂ 2 u3  f1
λ  2μ  μ  λ  μ ∂x∂z ∂t2 ∂x2 ∂z2 
∂2u ∂ 2 u2 ∂ 2 u2 ρ 22
μ  2  f2 ∂t ∂x2 ∂z ρ

ρ

∂ 2 u3 ∂ 2 u1 ∂2u ∂2u  μ 23  λ  2μ 23  f3 ;
λ  μ 2 ∂x∂z ∂t ∂x ∂z

(29) where λ and μ are the Lamé constants.

3348

D. Yang, N. Wang, S. Chen, and G. Song

In the numerical experiment, we choose the medium constants λ
2:75 GPa and μ
6:25 GPa, and the density ρ
2:1 g=cm3 . The source is an explosive source that is at the center of the computational domain and has a Ricker wavelet with a peak frequency of f0
15 Hz. The time variation of the source function is f1
f2
f3
5:76f20 1  160:6f0 t  12  exp80:6f0 t  12 :

(30)

The receiver is located at Rx; z
5:85 km; 6:75 km; the rest of the parameters are the same as those mentioned previously. Figure 10 presents the waveforms of three components in the isotropic medium at the receiver R, generated by the IRK-DSM and the fourth-order LWC, respectively. From Figure 10 we can see that the IRK-DSM has less or no visible numerical dispersion, whereas the fourth-order LWC suffers from the numerical dispersion. VTI Model

To compute the displacement component u2 , we choose space and time steps of Δx
Δz
45 m and Δt
0:02 sec, corresponding to the Courant number of Δt
0:767 because of the S-wave velocity of α
c Δx 1:725 km=sec. For horizontal component u1 and vertical component u3 , we choose spatial and time increments by Δx
Δz
45 m and Δt
0:01 sec, resulting in the maximal Courant number of 0.599 because of the P-wave velocity of 2:695 km=sec. The computational domain is 0 < x ≤ 13:5 km and 0 < z ≤ 13:5 km; the number of grid points is 301 × 301. Figure 9 shows the wave-field snapshot at t
1:6 sec, generated by the IRK-DSM. From the three-component snapshots, we can see that the P and SV waves presented in the horizontal component u1 (Fig. 9a) and vertical component u3 (Fig. 9c) are very clear even when a coarse grid (Δx
Δz
45 m) is chosen. It indicates that our present method enables wave propagation to be simulated in largescale models through using the coarse computation grids. For comparison, we present the waveforms computed using the IRK-DSM and the fourth-order LWC method.

Now we consider a two-layered VTI model, for which the elastic constants and densities in the upper and lower layers are listed in Table 3. The spatial and time increments are chosen by Δx
Δz
20 m and Δt
0:0092 sec for computing the component u2 , and Δt
0:003 sec for computing both the horizontal component u1 and the vertical component u3 , respectively. The computational domain is 0 < x, z ≤ 6 km. The source is located at xs ; zs 
3 km; 2:88 km. The rest of the computational parameters and the computer environment are the same as those in the isotropic model. The wave-field snapshots of the three components for u1 , u2 , and u3 at time 0.96 sec are presented in Figure 11. Figure 11 illustrates that the IRK-DSM has quite less or nearly no numerical dispersion and source-generated noises (artifacts due to source location at grid points), even for large vertical velocity contrast that reaches about 46.4% calculated from Table 3. Figure 11 also shows numerous reflected, transmitted, and converted phases. We can see from Figure 11b that the wavefront of the qSH-wave is an ellipse and the qP- and qSV-waves (Figs. 11a,c) show the directional dependence on the propagation velocity. The qSV wavefronts can have cusps and triplications depending on the value of C13 (Faria and Stoffa, 1994). Triplications can be observed in the horizontal component qSV-wavefront (shown in Fig. 11a) and in the vertical component qSVwavefront presented in Figure 11c. Moreover, we can also observe that the arrival times of qSH- and qSV-waves

Figure 9. Snapshots of seismic-wave fields for three components at time 1.6 sec in an isotropic medium, generated by the IRK-DSM, for (a) u1 component, (b) u2 component, and (c) u3 component.

Figure 10.

A comparison of three-component waveforms in a homogeneous isotropic medium. (a), (b) The synthetic seismograms are generated by the IRK-DSM and the fourth-order LWC method, respectively.

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations Table 3 Medium Parameters Used in the VTI Model Layer (No.)

Thickness

C11

C13

C33

C44

C66

ρs g cm3

1 2

3000 3000

14.2 40.8

5.4 13.2

18 50.6

6.5 25

3.8 13.8

3.2 4.2

are different through comparing Figure 11b with Figures 11a and c.

Conclusion and Discussion In this article, we propose an explicit IRK-DSM via the implicit Runge–Kutta method and truncated differentiator series method to solve the acoustic- and elastic-wave equations. We first transform the wave equation into a system of ordinary differential equations, and then use an implicit Runge–Kutta method associated with the differentiator series method to solve the ODEs. Specifically, we first transform the original wave equation (2) into equation (8). Then we use the interpolation approximations (A3)–(A9) to approximate the high-order spatial derivatives on the right-hand side of equation (8), converting equation (8) into a system of semidiscrete ordinary differential equations (ODEs). Finally, on the basis of both the implicit RK method and the differentiator series method, we develop the explicit IRK-DSM using both the truncated difference-operator Taylor expansion and the split-operator method. The error analysis shows that the IRK-DSM is third-order accuracy in time and fourth-order accuracy in space; the numerical error is less than those of the fourth-order LWC method and second-order FDM under the same condition.

Figure 11. Snapshots of seismic-wave fields for three components at time 0.96 sec on the spatial step Δx
Δz
20 m for (a) u1 component, (b) u2 component, and (c) u3 component.

3349

We also obtain the stability criteria of the IRK-DSM for solving 1D and 2D scalar wave equations, and the maximal Courant number αmax of the IRK-DSM are 0.968 and 0.844, respectively, which are slightly larger than those of the fourth-order and eighth-order LWC methods as shown in Table 2 because of combing the implicit RK method with the truncated differentiator series. Numerical dispersion analysis shows that the numerical dispersion of the IRK-DSM decreases as the Courant number increases, and the numerical dispersion reaches its minimum when the Courant number α equals the maximum Courant number αmax. So we can minimize the numerical dispersion of the IRK-DSM by choosing an optimal Courant number that approximates or equals the maximum Courant number αmax in practical seismic-wave modeling. Generally speaking, the numerical dispersion of the IRK-DSM is smaller than that of the fourth-order and eighth-order LWC methods (Dablain, 1986) under the same Courant number (see Figs. 4, 5, and 6) especially in the high-frequency range. So, when the velocity and spatial increments are constants, through increasing the temporal increment under keeping the IRK-DSM stable, we can not only lessen the numerical dispersion, but we can also reduce the computational cost and save the storage for computer codes. We have noticed previously that the CPU time of the IRK-DSM is more expensive than that of the fourth-order and eighth-order LWC methods (Dablain, 1986) under the condition of the same grid interval and time step. However, because the IRK-DSM yields less numerical dispersion than that of the high-order finite-difference methods such as highorder LWC methods when big grid is used, we can reduce the computational cost and storage space through using a larger time increment and a coarse spatial grid for a fixed-size model to achieve the same dispersion accuracy as those of the high-order LWC methods on a finer spatial grid with smaller time increment for suppressing the numerical dispersion. As is confirmed in the numerical dispersion section, the computational speed of the IRK-DSM when big grid is used is roughly 51 times that of the fourth-order LWC and about 17 times that of the eighth-order LWC method on a fine grid to achieve the same dispersion accuracy of the IRK-DSM; the space storage of the IRK-DSM requires only roughly 12.6% of the fourth-order LWC method and about 28.2% of that of the eighth-order LWC method, respectively. In this IRK-DSM, when determining these high-order spatial derivatives included in equation (8), the IRK-DSM uses not only the values of the displacement U and the particle velocity W at the mesh point (i, j) and its neighboring grid points (see equations [A3]–[A9]), but also the values of the gradients of the displacement U and the particle velocity W. Based on such a structure, the IRK-DSM retains more wave-field information included in the displacement function, the particle velocity, and their gradients. As a result, the IRK-DSM can effectively suppress the numerical dispersion and source-generated noises caused by discretizing the wave equations when too-coarse grids are used or when the models have a large velocity contrast between

3350

D. Yang, N. Wang, S. Chen, and G. Song

adjacent layers. Meanwhile, the IRK-DSM has higher spatial accuracy though the method only uses a local differenceoperator to approximate the high-order partial differential operator included in wave equations, in which three grid points are used in a spatial direction. Besides, this IRK-DSM provides more extra seismic information because it simultaneously computes displacement field, its time derivatives, and their spatial gradients. We initiate possible, more complicated applications of the IRK-DSM in large-scale seismic modeling, reverse time migration, and inversion based on the wave equations, for which the computation time and memory requirements are the bottleneck for their vast applications.

Data and Resources All data used in this article were produced synthetically.

Acknowledgments We thank Fred F. Pollitz (associate editor) and one anonymous reviewer for their detailed comments and helpful suggestions that greatly contributed to improving the article. Special thanks to Biaolong Hua and Jiming Peng for their revisions that substantially improved the presentation of the article. This work was supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 40725012) and the National Major Fundamental Research and Development Project of China (No. 2007CB411704).

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Igel, H., P. Mora, and B. Riollet (1995). Anisotropic wave propagation through finite-difference grids, Geophysics 60, 1203–1216. Johnson, C. (1990). Adaptive finite element methods for diffusion and convection problems, Comput. Meth. Appl. Mech. Eng. 82, 301–322. Ke, H. L. (1996). Calculation of generalized integrals using differentiator series method, J. Chongqing Jianzhu Univ. 18, 110–113 (in Chinese). Ke, H. L., and H. X. Xie (1999). Differentiator series solution of linear differential ordinary equation, Appl. Math. Mech. 20, 59–66. Kelly, K., R. Ward, S. Treitel, and R. Alford (1976). Synthetic seismograms: A finite-difference approach, Geophysics 41, 2–27. Komatitsch, D., and J.-P. Vilotte (1998). The Spectral Element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seismol. Soc. Am. 88, 368–392. Komatitsch, D., C. Barnes, and J. Tromp (2000). Simulation of anisotropic wave propagation based upon a spectral-element method, Geophysics 65, 1251–1260. Kondoh, Y., Y. Hosaka, and K. Ishii (1994). Kernel optimum nearly analytical discretization algorithm applied to parabolic and hyperbolic equations, Comput. Math. Appl. 27, 59–90. Kosloff, D., and E. Baysal (1982). Forward modeling by a Fourier method, Geophysics 47, 1402–1412. Kosloff, D., M. Reshef, and D. Loewenthal (1984). Elastic wave calculations by the Fourier method, Bull. Seismol. Soc. Am. 74, 875–891. Lele, S. K. (1992). Compact finite difference scheme with spectral-like resolution, J. Comp. Phys. 103,16–42. Levander, A. R. (1988). Fourth-order finite-difference P-SV seismograms, Geophysics 53, 1425–1436. Priolo, E., and G. Seriani (1991). A numerical investigation of Chebyshev spectral element method for acoustic wave propagation, Proc. of the 13th World Congress on Computation and Applied Mathematics, Vol. 2, 551–556. Priolo, E., J. M. Carcione, and G. Seriani (1994). Numerical simulation of interface waves by high-order spectral modeling techniques, J. Acoust. Soc. Am. 95, 681–693. Richtmyer, R. D., and K. W. Morton (1967). Difference Methods for Initial Value Problems, Interscience, New York. Sei, A., and W. Symes (1994). Dispersion analysis of numerical wave propagation and its computational consequences, J. Sci. Comput. 10, 1–27. Takeuchi, N., and R. J. Geller (2000). Optimally accurate second order timedomain finite difference scheme for computing synthetic seismograms in 2D and 3D media, Phys. Earth Planet. Interiors 119, 99–131. Turner, M. J., R. W. Clough, H. C. Martin, and L. J. Topp (1956). Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23, 805–823. Virieux, J. (1986). P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics 51, 889–901. Vlastos, S., E. Liu, I. G. Main, and X. Y. Li (2003). Numerical simulation of wave propagation in media with discrete distributions of fractures: Effects of fracture sizes and spatial distributions, Geophys. J. Int. 152, 649–668. Wang, S. Q., D. H. Yang, and K. D. Yang (2002). Compact finite difference scheme for elastic equations, J. Tsinghua Univ. (Sci. & Tech.) 42, 1128–1131 (in Chinese). Whiteman, J. R. (1975). A Bibliography for Finite Elements, Academic Press, New York. Yang, D. H., E. Liu, and Z. J. Zhang (2008). Evaluation of the u  W finite element method in anisotropic porous media, J. Seism. Expl. 17, 273–299. Yang, D. H., E. Liu, Z. J. Zhang, and J. Teng (2002). Finite-difference modeling in two-dimensional anisotropic media using a flux-corrected transport technique, Geophys. J. Int. 148, 320–328. Yang, D. H., J. M. Peng, M. Lu, and T. Terlaky (2006). Optimal nearly analytic discrete approximation to the scalar wave equation, Bull. Seismol. Soc. Am. 96, 1114–1130.

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations Yang, D. H., G. J. Song, S. Chen, and B. Y. Hou (2007). An improved nearly analytical discrete method: An efficient tool to simulate the seismic response of 2D porous structures, J. Geophys. Eng. 4, 40–52. Yang, D. H., J. W. Teng, Z. J. Zhang, and E. Liu (2003). A nearly-analytic discrete method for acoustic and elastic wave equations in anisotropic media, Bull. Seismol. Soc. Am. 93, 882–890. Zahradnik, J., P. Moczo, and T. Hron (1993). Testing four elastic finitedifference schemes for behavior at discontinuities, Bull. Seismol. Soc. Am. 83, 107–129. Zhang, Z. J., G. J. Wang, and J. M. Harris (1999). Multi-component wavefield simulation in viscous extensively dilatancy anisotropic media, Phys. Earth Planet. Interiors 114, 25–38. Zheng, H. S., Z. J. Zhang, and E. R. Liu (2006). Nonlinear seismic wave propagation in anisotropic media using the flux-corrected transport technique, Geophys. J. Int. 165, 943–956.

1 1 n n E1  E1 E1  E1 x ∂z U i;j  z ∂x Ui;j 2Δx x 2Δz z 1 1 1 1 1 n E1 E1  E1x E1  z  Ex Ez  Ex Ez Ui;j ; 4ΔxΔz x z

∂xz Uni;j


(A5) ∂3x V ni;j


∂3z V ni;j


Appendix A ∂2xz V ni;j


Evaluation of High-Order Derivatives When we use the explicit IRK-DSM to compute the values of U at time tn1 in synthetic seismograms, we first need the high-order space derivatives included in equations (9), (16), and (17), or equation (8). To determine the high-order derivatives, following Konddoh et al. (1994) and Yang et al. (2003), we introduce the interpolation function of the spatial steps Δx and Δz as follows
 5 X 1 ∂ ∂ r Δx  Δz V; (A1) GΔx; Δz
r! ∂x ∂z r
0 and construct the connection relations between the point (i, j) and its neighboring nodes. For example, at the grid point (i  1, j) we have the following connection relations, GΔx; 0ni;j
V ni1;j ; n n
∂ ∂ GΔx; 0 V
; ∂Δx ∂x i;j i1;j  n n
∂ ∂ GΔx; 0 V
∂Δz ∂z i;j i1;j



(A2)

and similarly the remaining 21 connection relations at the other 7 neighboring nodes can be easily written. Notice that the vector V is defined by V
U; WT, where U and W are the displacement and the particle velocity, respectively. From the 24 relations at the nodes (i  1, j), (i  1, j), (i, j  1), (i, j  1), (i  1, j  1), (i  1, j  1), (i  1, j  1), and (i  1, j  1), we can obtain the following approximation formulae, which are the similar expressions to those suggested by Yang et al. (2003, 2007). For convenience, we present the approximation formulae as follows ∂2x V ni;j

2 2 n 1 n E1  E1
δ x V i;j  x ∂x V i;j ; 2Δx x Δx2

(A3)

2 2 n 1 n E1  E1 δ V  z ∂z V i;j ; 2 z i;j 2Δz z Δz

(A4)

∂2z V ni;j


3351

15 n E1x  E1 x V i;j 2Δx3 3 n  E1x  8I  E1 x ∂x V i;j ; 2Δx2 15 n E1z  E1 z V i;j 2Δz3 3 n  E1z  8I  E1 z ∂z V i;j ; 2Δz2

(A6)

(A7)

1 1 1 1 1 1 5E1x E1z  5E1 x Ez  Ex Ez  Ex Ez 4Δx2 Δz 1 1 n  4E1z  4E1 z  6Ex  6Ex V i;j 1 1 1 1 E1x E1z  E1 x Ez  Ex  Ex 2ΔxΔz 1 2  2δ2z ∂x V ni;j  δ x ∂z V ni;j ; (A8) Δx2 

∂x2z V ni;j


1 1 1 1 1 1 5E1x E1z  5E1 x Ez  Ex Ez  Ex Ez 4Δx2 Δz 1 1 n  4E1x  4E1 x  6Ez  6Ez V i;j 1 1 1 1 E1x E1z  E1 x Ez  Ez  Ez 2ΔxΔz 1  2δ2x ∂z V ni;j  2 δ 2z ∂x V ni;j ; (A9) Δz 

where δ2z V ni;j
V ni;j1  2V ni;j  V ni;j1 , E1z V ni;j
V ni;j1 , n n 2 1 1 can be E1 z V i;j
V i;j1 . The operators δ x , Ex , and Ex n n n similarly defined. And V i;j , ∂x V i;j , ∂z V i;j , and ∂mxkz V ni;j ∂ ∂ denote ViΔx; jΔz; nΔt, ∂x ViΔx; jΔz; nΔt, ∂z ViΔx; mk m k n V=∂x ∂z i;j , respectively. jΔz; nΔt, and ∂

Appendix B Derivation of Stability Criteria 1D Homogeneous Case For the 1D case, these formulae (A3) and (A6) can be degenerated into the following formulae: ∂2x V nj


2 2 n 1 n E1  E1 δx V j  x ∂x V j ; 2Δx x Δx2

∂3x V nj


15 n E1x  E1 x V j 2Δx3 3 n  E1x  8I  E1 x ∂x V j : 2Δx2

(B1)

(B2)

3352

D. Yang, N. Wang, S. Chen, and G. Song

To obtain the stability condition of the IRK-DSM, we consider the harmonic solution of equations (9), (16), and (17) for the 1D case. Substituting the following solution 0 n 1 u n C B w C (B3) V
nj
B @ ∂x un A expikjh; n ∂x w

g21
e3iθ α2 fα4 r3 1:375  2:75r  α4 e6iθ 1:375  2:75rr3  α2 eiθ r0:25  0:25r  27α2 r2  54α2 r3   α2 e5iθ r0:25  0:25r  27α2 r2  54α2 r3   e2iθ 2  α4 129:375  258:75rr3  α2 r16  16r  e4iθ 2  α4 129:375  258:75rr3  α2 r16  16r

into equations (9), (16), and (17) with relations (B1) and (B2), we can obtain the following equation V
n1
GV
n ; where the amplification matrix G is 2 g11 g12 g13 6 g21 g22 g23 G
6 4 g31 g32 g33 g41 g42 g43

 e3iθ 4  α2 31:5  31:5rr  α4 r3 310  620rg=Δt

(B4) 3 g14 g24 7 7 g34 5 g44

g22
1  α2 2  eiθ  eiθ   α4 e2iθ r2 0:375  e2iθ 47:25  94:5r  e4iθ 0:375  0:75r

(B5)

 0:75r  eiθ 24  48r  e3iθ 24  48r  α6 e3iθ r4 0:6875  e2iθ 64:6875  129:375r

with g11
1  α 2  e 2

iθ

iθ

 e4iθ 64:6875  129:375r  eiθ 13:5  27r

4 2iθ 2

e α e

r 0:375

 e5iθ 13:5  27r  1:375r

 e 47:25  94:5r  e 0:375  0:75r  0:75r 2iθ

4iθ

 e6iθ 0:6875  1:375r

 eiθ 24  48r  e3iθ 24  48r

 e3iθ 155  310r

 α6 e3iθ r4 0:6875  e2iθ 64:6875  129:375r  e4iθ 64:6875  129:375r  eiθ 13:5  27r g23


 e6iθ 0:6875  1:375r  e3iθ 155  310r

(B6) g12
e2iθ fα4 0:25  0:5rr3  α4 e4iθ 0:25  0:5rr3  α2 eiθ r2  2r  16α2 r2  32α2 r3   α2 e3iθ r2  2r  16α2 r2  32α2 r3   e2iθ 1  α4 31:5  63rr3  α2 r4  4rgΔt

(B7) iθ

iθ

g13
α 0:25e h  0:25e

h  α h12e 4

(B11)



 e5iθ 13:5  27r  1:375r

2

(B10)

1 8eiθ  8eiθ  0:25e2iθ  0:25e2iθ r Δt  1 iθ iθ 2iθ 2iθ 2 8e  8e  0:25e  0:25e r   Δt  1 111:25eiθ  111:258eiθ  e2iθ  e2iθ  α6 h Δt 1 222:5eiθ  0:25e3iθ  0:25e3iθ r3   Δt α4 h

iθ

 222:5e  2e 

iθ

2iθ

 2e

2iθ

3iθ

 0:5e

  0:5e r 

α2 eiθ 0:5  0:5e2iθ h Δt

3iθ

4

(B12)

 12eiθ  0:375e2iθ  0:375e2iθ r2  24eiθ  24eiθ  0:75e2iθ  0:75e2iθ r3   α6 h55:625eiθ  55:625eiθ  0:5e2iθ  0:5e2iθ  0:125e3iθ

 α4 h12eiθ  12eiθ

 0:125e3iθ r4  111:25eiθ  111:25eiθ  e2iθ  e2iθ  0:25e3iθ  0:25e3iθ r5 

g24
α2 0:25eiθ h  0:25eiθ h

(B8)

 0:375e2iθ  0:375e2iθ r2  24eiθ  24eiθ  0:75e2iθ  0:75e2iθ r3   α6 h55:625eiθ  55:625eiθ  0:5e2iθ

g14
α2 eiθ h0:5  e2iθ 0:5  0:5rrΔt  α4 h8eiθ  8eiθ  0:25e2iθ  0:25e2iθ r3 Δt  16eiθ  16eiθ  0:5e2iθ  0:5e2iθ r4 Δt

 0:5e2iθ  0:125e3iθ  0:125e3iθ r4  111:25eiθ  111:25eiθ  e2iθ

(B9)

 e2iθ  0:25e3iθ  0:25e3iθ r5 

(B13)

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

g31


3:75eiθ  3:75eiθ 1  fα4 180eiθ
α h h

g42
α2

2

3353


3:75eiθ  3:75eiθ 1  fα4 180eiθ h h

 180eiθ  5:625e2iθ  5:625e2iθ r2

 180eiθ  5:625e2iθ  5:625e2iθ r2

 360eiθ  360eiθ  11:25e2iθ  11:25e2iθ r3 g

 360eiθ  360eiθ  11:25e2iθ  11:25e2iθ r3 g

1  fα6 834:375eiθ  834:375eiθ  7:5e2iθ h  7:5e2iθ  1:875e3iθ  1:875e3iθ r4

1  fα6 834:375eiθ  834:375eiθ  7:5e2iθ h  7:5e2iθ  1:875e3iθ  1:875e3iθ r4

 1668:75eiθ  1668:75eiθ  15e2iθ  15e2iθ

 1668:75eiθ  1668:75eiθ  15e2iθ  15e2iθ

 3:75e3iθ  3:75e3iθ r5 g

 3:75e3iθ  3:75e3iθ r5 g

(B14)

eiθ r7:5  e2iθ 7:5  7:5r  7:5rΔt h 1 4  fα 120eiθ  120eiθ  3:75e2iθ h  3:75e2iθ r3 Δt  240eiθ  240eiθ  7:5e2iθ

g32
α2

 7:5e2iθ r4 Δtg

(B15)

g43


(B19)

1 3iθ 2 4 3 e α fα r 0:375  0:75r Δt  α4 e6iθ 0:375  0:75rr3  α2 eiθ r1:5  1:5r  24α2 r2  48α2 r3   α2 e5iθ r1:5  1:5r  24α2 r2  48α2 r3   e2iθ 1:5  α2 36  36rr  α4 r3 661:875  1323:75r

g33
1  α2 6  0:75eiθ  0:75eiθ 

 e4iθ 1:5  α2 36  36rr

 α4 e2iθ r2 2:25  e2iθ 234  468r

 α4 r3 661:875  1323:75r

 eiθ 54  108r  4:5r  e3iθ 54  108r

 e3iθ 12  α2 156  156rr

 e4iθ 2:25  4:5r  α6 e3iθ r4 0:1875

 α4 r3 2100  4200rg

 eiθ 12  24r  e5iθ 12  24r

(B20)

 e6iθ 0:1875  0:375r  0:375r  e2iθ 330:938  661:875r

g44
1  α2 6  0:75eiθ  0:75eiθ 

 e3iθ 1050  2100r  e4iθ 330:938  661:875r

 α4 e2iθ r2 2:25  e2iθ 234  468r

(B16)

 eiθ 54  108r  4:5r  e3iθ 54  108r  e4iθ 2:25  4:5r  α6 e3iθ r4 0:1875  eiθ 12  24r  e5iθ 12  24r

g34
e2iθ fα4 r3 1:5  3r  α4 e4iθ r3 1:5  3r  α2 eiθ r1:5  1:5r  36α2 r2  72α2 r3 

 e6iθ 0:1875  0:375r  0:375r

 α2 e3iθ r1:5  1:5r  36α2 r2  72α2 r3 

 e2iθ 330:938  661:875r

 e2iθ 1  α4 156  312rr3

 e3iθ 1050  2100r

 α2 r12  12rgΔt

g41


(B17)

α4

120eiθ  120eiθ  3:75e2iθ  3:75e2iθ r hΔt  120eiθ  120eiθ  3:75e2iθ  3:75e2iθ r2  α6 1668:75eiθ  1668:75eiθ  15e2iθ hΔt  15e2iθ  3:75e3iθ  3:75e3iθ r3  3337:5eiθ



 3337:5eiθ  30e2iθ  30e2iθ  7:5e3iθ  7:5e3iθ r4  

α2 eiθ 7:5  7:5e2iθ  hΔt

(B18)

 e4iθ 330:938  661:875r

(B21)

p 3 6 ,

in which θ
kh, h
Δx, r
12  the Courant number (Dablain, 1986; Sei and Symes, 1994) is defined by α
c0 Δt=h with h
Δx. Let G denote the conjugate transpose matrix of G. Following the analyses of Richtmyer and Morton (1967) and Guan and Lu (2006), we know that the IRK-DSM with the amplification matrix G is stable if the spectral radius ρG · G of the matrix G · G satisfies ρG · G ≤ 1. So we can obtain the stability condition, deriving from the condition of ρG · G ≤ 1, as follows α ≤ αmax ≤ 0:968;

(B22)

3354

D. Yang, N. Wang, S. Chen, and G. Song

0

or Δt ≤ αmax

h h ≈ 0:968 ; c0 c0

(B23)

where αmax denotes the maximum Courant number that keeps the numerical calculation stable.

1 u0 B w0 C C V
nj
B @ ∂x u0 A expiωnum nΔt  kjh ∂x w0

into equations (9), (16), and (17) with relations (B1) and (B2) to obtain the dispersion equation as follows: DetM
0;

2D Homogeneous Case For the 2D problem, we consider the case of Δx
Δz
h. Following the same steps as discussed in the 1D case, we can obtain the following stability condition Δt ≤ αmax

h h ≈ 0:844 : c0 c0

(B24)

Appendix C Derivation of the Numerical Dispersion Relation To minimize the dispersion error, we derive the numerical dispersion relation of the IRK-DSM for the 1D case. For this, following the dispersion analysis methods presented in Dablain (1986) and Yang et al. (2006), we consider the harmonic solution of equation (9) and substitute the solution

(C1)

(C2)

where k is the wavenumber. Owing to the complexity of elements of the matrix M, we omit the detail expressions of the matrix M. From the dispersion relation (C2), we can obtain the ratio of the numerical velocity to the phase velocity c0 as follows: R


cnum ωnum Δt γ
;
c0 αθ αθ

(C3)

where α is the Courant number, θ
kh, h
Δx, and γ
ωnum Δt satisfy the dispersion equation (C3).

Department of Mathematical Sciences Tsinghua University Beijing 100084, China [email protected]

Manuscript received 30 November 2008