High-order time discretizations in seismic modeling

Transcription

1 GEOPHYSICS, VOL. 72, NO. 5 SEPTEMBER-OCTOBER 27; P. SM115 SM2, 5 FIGS / High-order time dicretization in eimic modeling Jing-Bo Chen 1 ABSTRACT Seimic modeling play an important role in exploration geophyic. High-order modeling cheme are in demand for practical reaon. In thi context, I preent three kind of high-order time dicretization: Lax-Wendroff method, Nytröm method, and plitting method. Lax- Wendroff method are baed on the Taylor expanion and the replacement of high-order temporal derivative by patial derivative, Nytröm method are implified Runge-Kutta algorithm, and plitting method comprie ubtep for onetep computation. Baed on thee method, three cheme with third-order and fourth-order accuracy in time and peudopectral dicretization in pace are preented. I alo compare their accuracy, tability, and computational complexity, and dicu advantage and hortcoming of thee algorithm. Numerical experiment how that the fourth-order Lax-Wendroff cheme i more efficient for hort-time imulation while the fourth-order Nytröm cheme and the thirdorder plitting cheme are more efficient for long-term computation. INTRODUCTION Seimic modeling i an important foundation of eimology. Baed on the calar wave equation or elatic wave equation, eimic modeling can produce ynthetic eimogram, which play a key role in eimic interpretation and exploration. Seimic modeling method can be claified into three main categorie: direct method, integral-equation method, and ray-tracing method. Each method category ha it advantage and hortcoming. A good review of thee method i given by Carcione et al. 22. High-accuracy eimic modeling cheme become more important a computing capacity increae. Numerical cheme for direct method of eimic modeling involve dicretization of both pace and time variable. Spatial dicretization have been explored extenively, and three approache have been developed: finite-difference method Alford et al., 1974; Kelly et al., 1976, peudopectral method Gazdag, 1981; Koloff and Bayal, 1982, and finite-element method Marfurt, Some combination of thee method are alo available, uch a pectral-element method Komatitch and Vilotte, 1998 and finite-volume method Dormy and Tarantola, On the other hand, temporal dicretization are relatively le tudied. Mot of the numerical cheme ue econd-order finite-difference time dicretization. However, for typical time-tep ize, time dicretization exhibit evere diperion error for long-time imulation Chen, 26. Current tudie in eimology uually need large-cale and long-time eimic modeling; therefore, high-order time dicretization are in demand. Three kind of approache for high-order time dicretization are available: Lax-Wendroff method, Nytröm method, and plitting method. Lax-Wendroff method ue patial derivative to replace high-order time derivative Dablain, Nytröm method are implified Runge-Kutta algorithm that take advantage of the pecial form of the wave equation Hairer et al., 1993; Chen, 26. Baed on vector-field decompoition, plitting method conit of everal ubtep for one computational tep Qin and Zhang, 199; Yohida, 199. In the next ection, uing peudopectral patial dicretization, I preent the three method for time dicretization. Thi i followed by an analyi of their accuracy, tability, and computational complexity. I then perform numerical experiment to demontrate the preented algorithm. HIGH-ORDER TIME DISCRETIZATIONS Seimic modeling i baed on the calar wave equation or the elatic wave equation. Becaue I am dealing mainly with time dicretization, I will ue the calar wave equation in two patial dimenion for the ake of implicity. For the cae of an elatic wave equation or for three patial dimenion, imilar reult can be obtained in a imilar way. For the patial dicretization, I ue peudopectral method for which error mainly arie from the time dicretization. Peudopectral method for modeling the calar wave equation are preented by Gazdag The main point of peudopectral method Manucript received by the Editor October 29, 26; revied manucript receivedapril 16, 27; publihed onlineaugut 23, ChineeAcademy of Science, Intitute of Geology and Geophyic, Beijing, China. chenjb@vip.ohu.com. 27 Society of Exploration Geophyicit. All right reerved. SM115

2 SM116 Chen i to expre the wavefield under conideration in term of a complete et of orthogonal bai function whoe derivative are known exactly. In practice, Fourier peudopectral method are uually employed. Conider the calar wave equation, c2 2 u t 2 = 2 u x u z 2, where ux,z,t i the wavefield and cx,z i the velocity. Let u = u 1,1,...,u Nx,N z T, where T repreent the tranpoe and u i,l are the wavefield value at dicrete location, i.e., u i,l uix,lz,t, i = 1,...,N x ; l = 1,...,N z. Here, x and z are grid increment in the x- and z-direction, and N x and N z are the number of grid line in the x- and z-direction, repectively. The emidicrete equation reulting from the peudopectral method for equation 1 i 1 d 2 u dt 2 = c2 F 1 w * Fu, 2 where F and F 1 repreent 2D forward and invere finite Fourier tranform, repectively, and w = w 1,1,...,w Nx,N z T with w i,l = k 2 xi + k 2 zl, where k xi and k zl are dicrete wavenumber in the x- and z-direction, repectively. The tar * denote array multiplication between vector. For example, uppoe that p = p 1,p 2,...,p m and q = q 1,q 2,...,q m, then p*q = p 1 q 1,p 2 q 2,...,p m q m. Lax-Wendroff method Baed on Taylor expanion, Lax-Wendroff method ue patial derivative to replace high-order time derivative Dablain, 1986; Carcione et al., 22: u n+1 2u n + u n 1 t 2 2 = c 2 u n x u n z 2 J +2 j =2 t 2j 2 2j! 2j u n t 2j, where u n ux,z,nt and t i the time-tep ize. The econd-order time derivative i provided by equation 1, and the higher-order time derivative are obtained by the following recurive formula: 2j u 2 n 2 t 2j = c x j 2 u n z t 2j 2. Scheme 3 ha an accuracy of Ot 2J. Taking J = 2, I obtain a fourth-order cheme: u n+1 2u n + u n 1 t 2 2 = c 2 u n x u n z 2 x z + c2 t 2 u 2c2 n x u n z 2. 5 Uing peudopectral patial dicretization, cheme 5 become u n+1 2u n + u n 1 t 2 = c 2 F 1 w * Fu n + c2 t 2 F 1 w * Fc 2 F 1 w * Fu n, where the additional tarting value u n can be obtained through the following Taylor expanion: 4 unt = i = i un 1t t i 6 t i. 7 i! Here, the high-order time derivative are obtained in the ame way a in cheme 3. Nytröm method The baic idea of Nytröm method i a follow: Firt, introduce a new variable to reduce the econd-order wave equation to an equivalent firt-order ytem; then apply Runge-Kutta method to the firt-order ytem and implify by taking advantage of the pecial form of the firt-order ytem Nytröm, 1925; Hairer et al., A Nytröm method for equation 2 read V i = c 2 F 1w * Fu n + d i tv n + t 2 a ij V j, i = 1,2,...,, u n+1 = u n + tv n + t 2 b iv i, v n+1 = v n + t b i V i, where v = du/dt, u n unt, v n vnt, u n+1 un +1t, v n+1 vn +1t, and d i, a ij, b i, and b i are contant. Here, refer to the number of the auxiliary variable V i in equation 8. Now I recall the definition about cheme order Hairer et al., Denote z = u,v T. Let zn +1t = un +1t,vn +1t T be the true olution to equation 2 and z n+1 = u n+1,v n+1 T be the numerical olution to equation 2 obtained with cheme 8. If the following inequality hold, j =1 zn +1t z n+1 Kt p+1, then cheme 8 ha order p. Here,. denote a norm and K i a contant. To obtain a cheme of order p, I calculate the Taylor erie for both zn +1t and z n+1 and make the two Taylor erie coincide up to the term t p. Thi procedure reult in algebraic equation atified by d i, a ij, b i, and b i, which are called order condition. The number of algebraic equation and therefore the coefficient increae with the order. In deriving the Taylor erie, I need to calculate the highorder derivative of the compoite function. Uually, it i very complicated and troubleome to calculate the high-order derivative directly. By etablihing a relationhip between derivative and their graphical repreentation tree, the order condition can be derived 8 9

3 High-order time dicretization SM117 eaily. By olving thee algebraic equation, I can obtain the correponding method. For detail, ee Hairer et al I alo conider a fourth-order explicit Nytröm method Qin and Zhu, 1991: where V 1 = c 2 F 1 w * Fu n + d 1 tv n, V 2 = c 2 F 1 w * Fu n + d 2 tv n + a 21 t 2 V 1, V 3 = c 2 F 1 w * Fu n + d 3 tv n + a 31 t 2 V 1 + a 32 t 2 V 2, u n +1 = u n + tv n + t 2 b 1V 1 + b 2V 2 + b 3V 3, v n +1 = v n + tb 1 V 1 + b 2 V 2 + b 3 V 3, d 1 = b 2 = 3+ 3, d 2 = b 1 = 5 3 3, 24, b 3 = b 2 = 1 2, b 3 = 3+2 3,, d 3 = 3+ 3, 6, b 1 = 3 2 3, 1 a 21 = 2 3 3, a 31 =, a 32 = Scheme 1 i an explicit algorithm; therefore, I have a 11 = a = a 13 = a 22 = a 23 = a 33 =. Splitting method Baed on vector-field decompoition, plitting method conit of everal ubtep for one-tep computation Yohida, 199; Qin and Zhang, 199.Aplitting method for equation 2 read a follow: v i = v i 1 + p i tc 2 F 1 w * Fu i 1, u i = u i 1 + q i tv i, i = 1,2,...,I, where the upercript i denote the intermediate reult and u = u n unt, v = v n vnt, u I = u n+1 un +1t, and v I = v n+1 vn +1t. Thi mean I need to compute I ubtep to obtain the reult for time level n + 1 from that of time level n. The coefficient p i and q i determine the accuracy of the plitting method. I take I = 3 and conider a third-order method Ruth, 1983: v 1 = v n + p 1 tc 2 F 1 w * Fu n, u 1 = u n + q 1 tv 1, v 2 = v 1 + p 2 tc 2 F 1 w * Fu 1, u 2 = u 1 + q 2 tv 2, v n+1 = v 2 + p 3 tc 2 F 1 w * Fu 2, u n+1 = u 2 + q 3 tv n+1, 13 where p 1 = 7/24, p 2 =3/4,p 3 = 1/24, q 1 = 2/3, q 2 = 2 3, q 3 =1. ACCURACY, STABILITY, AND COMPUTATIONAL COMPLEXITY In thi ection, I dicu the accuracy, tability, and computational complexity of cheme 6, 1, and 13. Accuracy and tability Both cheme 6 and 1 have a fourth-order accuracy in time: Ot 4. Scheme 13 ha a third-order accuracy in time: Ot 3. Becaue I ue peudopectral patial dicretization, the patial accuracy i of exponential order Oexpx Fornberg, Therefore, the total accuracy of cheme 6 and 1 i Ot 4 + expx; for cheme 13, it i Ot 3 + expx. I take x = z = h and denote that the Courant number a r = ct/h. Uing the tandard pectral analyi technique Koloff and Bayal, 1982, I can obtain the following tability limit for the three cheme: Computational complexity r.7797 for cheme 6, r.5826 for cheme 1, r.5645 for cheme The main computational cot in cheme 6, 1, and 13 come from the evaluation of pair of fat Fourier tranform FFT forward and invere. They involve 2, 3, and 3 pair of FFT, repectively. Therefore, the computational complexitie of cheme 6, 1, and 13 are 4N x N z log 2 N x N z,6n x N z log 2 N x N z, and 6N x N z log 2 N x N z, repectively. Scheme 6 ha le computational complexity, but it need evaluation of the additional tarting value through the Taylor expanion 7. Another hortcoming of cheme 6 i that it involve patial derivative of the velocity explicitly. Therefore, it i neceary to dicretize the patial derivative of the velocity, and thi dicretization may reult in ome problem for dicontinuou media Kelly et al., On the other hand, cheme 1 and 13 do not need the evaluation of the additional tarting value and involve no patial derivative of the velocity explicitly. Another advantage of cheme 1 and 13 i that they are ymplectic algorithm, which have good performance for long-time imulation Chen, 26. See Appendix A for an introduction to ymplectic algorithm. NUMERICAL EXPERIMENTS In thi ection, I perform numerical experiment to tet the preented cheme. Like Gazdag 1981, I ue the initial condition:

4 SM118 Chen Velocity (m/) d) Figure 1. a The velocity model, and wavefield at t = 1.86 computed with cheme 6b,1c, and 13d d) 5 6 Figure 2. Impule repone at time t = 1.8 computed with cheme 16a,6b,1c, and 13d. 5 6 ux,z, = exp a 2 x 2 + z z 2 and ux,z, =, 15 t where z i a contant that indicate the poition of the ource and a i a contant determining the width of the pule. A demontrated in Gazdag 1981, I et ax =.5 to guarantee the reflected ignal quality. In the experiment, I take x = z =5m,a =.1 and ue periodic boundary condition. The firt example i an inhomogeneou medium that conit of three region with velocitie of,, and m/, repectively. There are two interface: one i traight; the other i curved Figure 1a. The ource i et at x =m, z = 2 m. I take t =.6. Figure 1b d how the wavefield at t = 1.86 computed with cheme 6, 1, and 13, repectively. The wavefield conit of the incident wave, two reflected wave, and two tranmitted wave. The three reult have almot the ame performance, and they give a qualitatively correct imulation of reflection and tranmiion phenomena. The econd example compare cheme 6, 1, and 13. In thi cae, I chooe a homogeneou medium with a velocity of m/ for implicity and clarity. I alo conider a commonly ued econd-order cheme Gazdag, 1981: u n+1 2u n + u n 1 t 2 = c 2 F 1 w * Fu n. 16 Figure 2 how the impule repone at t = 1.8 computed by cheme 6, 1, 13, and 16. The ource i et at x =m,z = 32 m, and t =.6. The reult computed with cheme 16 tart to exhibit diperion error, although the other three reult uffer no uch error. Thi i becaue cheme 16 ha low accuracy in time. The CPU time for cheme 6, 1, and 13 i 3.93, 6.77, and 5.24, repectively, approximately agreeing with the computational complexity of the three cheme. The evaluation of the variable v in cheme 1 i more complex than that in cheme 13, although cheme 1 and 13 have the ame eential computational complexity. Figure 3 how the impule repone at t = 18. For thi computational time, the wavefront computed with cheme 16 ha blurred badly, although the wavefront computed with cheme 6, 1, and 13 are till harp. To reduce the diperion error computed with cheme 16, I can ue a maller time-tep ize. However, the overall computational cot therefore, CPU time of cheme 16 with a maller

5 High-order time dicretization SM119 time-tep ize will be much higher than cheme 1 with a larger time-tep ize for long-time imulation. Thi iue i dicued in detail in Chen 26. Now I perform a further comparion with the reult hown in Figure 2. For thi purpoe, I plot the amplitude curve at a fixed point x = 16 m, z = 16 m in Figure 4a. Note a dicrepancy between the amplitude curve computed with cheme 16 and the amplitude curve computed with the other cheme. However, the amplitude curve computed with cheme 6, 1, and 13 are inditinguihable in thi plot. From the enlarged portion of Figure 4a, the amplitude curve computed with cheme 6 blue curve can be ditinguihed from the amplitude curve computed with cheme 1 and 13 Figure 4b. To make careful comparion of the amplitude curve computed with the three cheme 6, 1, and 13, I plot their amplitude curve for a longer imulation time tepee Figure 4c and it enlarged portion Figure 4d. From Figure 4d, I ee that the three amplitude curve can be ditinguihed from one another. However, omething unuual happen. The amplitude curve computed by the third-order cheme 13 green curve and the fourth-order cheme 1 red curve are cloer to each other. According to the accuracy, the amplitude curve computed with fourth-order cheme 6 blue curve and fourth-order cheme 1 red curve hould have been cloer to each other. To explain thi phenomenon, I add an amplitude curve that i computed with cheme 6 with a maller time-tep ize t =.3 black curve; ee Figure 4e and it enlarged portion Figure 4f. In Figure 4f, the green curve i cloer to the black curve than the blue curve i. Thi indicate that for thi imulation time 6, the thirdorder cheme 13 i more accurate than the fourth-order cheme 6. I can explain thi a follow: Although the fourth-order cheme 6 i more accurate than the third-order cheme 13 for one time tep, the error growth of the former i fater than the later a time tep increae. Apoible explanation for the lower error growth of cheme 13 i that cheme 13 i a ymplectic algorithm and ymplectic algorithm have lower error growth than nonymplectic one ee Appendix A. Scheme 1 i alo a ymplectic algorithm Qin and Zhu, 1991; Chen, 26. The fourth-order cheme 1 i cloer to the black curve than the third-order cheme 13 becaue it ha greater accuracy. For thi imulation time 6 and imilar accuracy, cheme 1 with a time-tep ize of t =.6 i more efficient than cheme 6 with a maller time-tep ize of t =.3 becaue their total computational complexitie are and pair of FFT, repectively. In the final numerical example, I tet the two fourth-order cheme 6 and 1 on the Marmoui model Figure 5. In thi example, the initial condition are taken a ux,z, =, and ux,z, =. 17 t I employ a commonly ued Ricker wavelet with a peak frequency of 25 Hz Rehef et al., The Ricker wavelet i placed at x = 6 m and z = m. I et x =.5 m, z = 4 m, and t =.2. Figure 5 illutrate wavefield at t =.36 computed with thee two cheme. The two wavefield are almot the ame. However, making the ame comparion a in the homogeneou cae, one can ee that cheme 1 i more accurate than cheme 6 for the ame time-tep ize. To achieve the ame accuracy with thee two Figure 3. Impule repone at t = 18 computed with cheme 16a,6b,1c, and 13d d)

6 SM Figure 4. curve at a fixed point x = 16 m, z = 16 m. a curve computed with cheme 16 black curve, 6 blue curve,1red curve, and 13 green curve. b Enlarged portion of a. c curve computed with cheme 6 blue curve,1red curve, and 13 green curve for a longer imulation time 6. d Enlarged portion of c. e Additional amplitude curve computed with cheme 6 with a maller time-tep ize t =.3 added black curve. f Enlarged portion of e. Chen Time () Time () Time () e) Time () d) Time () f) Time () Depth (km) 1 2 Ditance (km) Velocity (m/) cheme, one mut ue a maller time-tep ize for cheme 6. For brevity, I have not hown the correponding plot, which are imilar to thoe in Figure 4. CONCLUSIONS Figure 5. a Marmoui model, and wavefield at t =.36 computed with cheme 6b and 1c. I have preented three modeling cheme with high-order time dicretization and patial peudopectral dicretization. The Lax- Wendroff cheme ha better tability but need evaluation of additional tarting value. It alo involve the patial derivative of velocity explicitly. The Nytröm cheme and the plitting cheme need no additional tarting value and involve no patial derivative of velocity explicitly. For hort-time imulation, the fourth-order Lax- Wendroff cheme i more efficient. For long-time imulation, the fourth-order Nytröm cheme and the third-order plitting cheme are more efficient. ACKNOWLEDGMENTS I would like to thank the anonymou reviewer for valuable uggetion and correction. Thi work i upported by the National Natural Science Foundation of China under Grant

7 High-order time dicretization SM1 APPENDIX A SYMPLETIC ALGORITHMS FOR HAMILTONIAN SYSTEMS In thi appendix, I preent the ymplectic algorithm for Hamiltonian ytem. For detail, ee Sanz-Serna and Calvo 1994 and Hairer et al. 22. A Hamiltonian ytem read dp dt dq dt = Hp,q, q = Hp,q, A-1 p where p = p 1,p 2,...,p n i the generalized coordinate, q = q 1,q 2,...,q n i the generalized momentum, and Hp,q i the Hamiltonian function. Suppoe that the olution to the Hamiltonian ytem A-1 i Z = FZ, where Z = pt,qt T and Z = p,q T. Here, T repreent the tranpoe of matrice. The olution Z = FZ atifie F Z J F T Z = J, where J = I I. A-2 Here, F/Z i the Jacobian of the vector-valued function FZ and I i the nn identity matrix. A function atifying the above equality i called a ymplectic mapping. Therefore, the true olution of a Hamiltonian ytem i a ymplectic mapping. A numerical method for Hamiltonian ytem i called a ymplectic algorithm if the reulting numerical olution i alo a ymplectic mapping. The numerical olution obtained by a ymplectic algorithm exactly atifie a perturbed Hamiltonian ytem. Thi property guarantee that ymplectic algorithm have lower error growth and poee remarkable capability in preerving conervative quantitie. Therefore, ymplectic algorithm play an important role in high-accuracy or long-time numerical imulation. In the following, I preent three kind of ymplectic algorithm. Symplectic Runge-Kutta method I apply a Runge-Kutta method to the Hamiltonian ytem A-1 and obtain P i = p n t j =1 Q i = q n + t j =1 p n+1 = p n t q n+1 = q n + t If the coefficient in cheme A-3 atify a ij q HP j,q j, a ij p HP j,q j, b i q HP i,q i, b i p HP i,q i. A-3 b i a ij + b j a ji b i b j =, i, j = 1,...,, A-4 then cheme A-3 i a ymplectic algorithm. Symplectic Nytröm method Scheme A-3 i implicit. If the Hamiltonian function in ytem A-1 ha the pecial form Hp,q = 1 2 ppt + Vq, A-5 then I can obtain explicit ymplectic Nytröm method. A Nytröm method for ytem A-1 read Q i = q V q n + d i tp n + t 2 a ij Q j, i = 1,2,...,, q n+1 = q n + tp n + t 2 b iq i, p n+1 = p n + t b i Q i. If the coefficient in cheme A-6 atify b i = b i 1 d i, j =1 i = 1,...,, A-6 b i b j a ij = b j b i a ji, i, j = 1,...,, A-7 then cheme A-6 i a ymplectic algorithm. Baed on the above criterion, one can how that cheme 1 i a ymplectic algorithm. Symplectic plitting method If the Hamiltonian function in ytem A-1 ha the eparable form Hp,q = Fp + Gq, A-8 one can obtain ymplectic plitting method. A ymplectic plitting method for ytem A-1 read p i = p i 1 e i t q Gqi 1, q i = q i 1 + d i t p Fpi, i = 1,2,...,I, A-9 where the upercript i denote the intermediate reult and where p = p n, q = q n, p I = p n+1, and q I = q n+1. I lit ome ymplectic plitting method a follow: Second order: =2e 1 = e 2 = 1 2, d 1 =1, d 2 =; Third order: =3e 1 = 7 24, e 2 = 3 4, e 3 = 1 24, d 1 = 2 3, d 2 = 2 3, d 3 =1;

8 SM2 Chen Fourth order: =4e 1 = e 4 = d 1 = d 3 = /3, e 2 = e 3 = 1 21/ /3, /3, d 2 = 2 1/ /3, d 4 =. Introducing a new variable v = du/dt, I can cat the econd-order equation 2 into a Hamiltonian ytem Chen, 26. Therefore, I can develop the correponding ymplectic algorithm. REFERENCES Alford, R. M., K. R. Kelly, and D. M. Boore, 1974, Accuracy of finite-difference modeling of the acoutic wave equation: Geophyic, 39, Carcione, J. M., G. C. Herman, and A. P. E. ten Kroode, 22, Seimic modeling: Geophyic, 67, Chen, J. B., 26, Modeling the calar wave equation with Nytröm method: Geophyic, 71, no. 5, T151 T158. Dablain, M. A., 1986, The application of high-order differencing to the calar wave equation: Geophyic, 51, Dormy, E., and A. Tarantola, 1995, Numerical imulation of elatic wave propagation uing a finite volume method: Journal of Geophyical Reearch, 1, Fornberg, B., 1996, Apractical guide to peudopectral method: Cambridge Univerity Pre. Gazdag, J., 1981, Modeling of the acoutic wave equation with tranform method: Geophyic, 46, Hairer, E., C. Lubich, and G. Wanner, 22, Geometric numerical integration: Springer-Verlag. Hairer, E., S. P. Nøett, and G. Wanner, 1993, Solving ordinary differential equation I: Springer-Verlag. Kelly, K. R., R. W. Ward, S. Treitel, and R. M. Alford, 1976, Synthetic eimogram: Afinite-difference approach: Geophyic, 41, Komatitch, D., and J. P. Vilotte, 1998, The pectral element method: An efficient tool to imulate the eimic repone of 2D and 3D geological tructure: Bulletin of the Seimological Society ofamerica, 88, Koloff, D., and E. Bayal, 1982, Forward modeling by the Fourier method: Geophyic, 47, Marfurt, K. J., 1984, Accuracy of finite-difference and finite-element modeling of the calar and elatic wave equation: Geophyic, 49, Nytröm, E. J., 1925, Üeber die numeriche Integration von Differentialgleichungen: Acta Societe Scientiarum Fennicae, 5, Qin, M. Z., and M. Q. Zhang, 199, Multi-tage ymplectic cheme of two kind of Hamiltonian ytem for wave equation: Computer and Mathematic withapplication, 19, Qin, M. Z., and W. J. Zhu, 1991, Canonical Runge-Kutta-Nytröm method for econd order ODE : Computer and Mathematic with Application, 22, Rehef, M., D. Koloff, M. Edward, and C. Hiung, 1988, Three-dimenional acoutic modeling by the Fourier method: Geophyic, 53, Ruth, R. D., 1983, A canonical integration technique: IEEE Tranaction on Nuclear Science, 3, Sanz-Serna, J. M., and M. Calvo, 1994, Numerical Hamiltonian problem: Chapman and Hall. Yohida, H., 199, Contruction of higher order ymplectic integrator: Phyic LetterA, 15,