Abstract：In this paper,we use the least square and Lagrange multiplier to perform difference coefficient optimization for the staggered compact finite difference scheme of first-order derivatives based on the idea of dispersion relationship preservation.Then,we analyze the optimized-scheme simulation accuracy,dispersion relation,and acoustic wave equation stability.The following understandings are obtained:A.For the same difference accuracy,the optimized staggered compact difference (OSCD) scheme uses two more nodes to calculate the first-order derivative than staggered compact difference (SCD) schemes;B.OSCD has a smaller truncation error and lower simulation dispersion than staggered difference (SD) schemes and SCD,so OSCD has higher calculation accuracy,and it is more suitable for coarse grid computing;C.For the same difference accuracy,the stability condition of OSCD for acoustic wave equation is slightly stricter than SD and SCD,and the applicable time grid size is slightly smaller.Wavefield numerical simulations of acoustic wave equation on uniform,horizontal-layered,and Marmousi models demonstrate that the proposed method is suitable for complex media and it has higher accuracy and efficiency.
唐佳,王凡,刘福烈.三维波动方程正演的三级并行加速[J].石油地球物理勘探,2016,51(5):1049-1054.TANG Jia,WANG Fan,LIU Fulie.3D wave equation forward modeling based on three-level parallel acce-leration[J].Oil Geophysical Prospecting,2016,51(5):1049-1054.
[2]
唐怀谷,何兵寿.一阶声波方程时间四阶精度差分格式的伪谱法求解[J].石油地球物理勘探,2017,52(1):71-80.TANG Huaigu,HE Bingshou.Pseudo spectrum method of first-order acoustic wave equation finite-difference schemes with fourth-order time difference accuracy[J].Oil Geophysical Prospecting,2017,52(1):71-80.
[3]
董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,2000,43(6):856-864.DONG Liangguo,MA Zaitian,CAO Jingzhong.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation[J].Chinese Journal of Geophysics,2000,43(6):856-864.
[4]
杜启振,郭成锋,公绪飞.VTI介质纯P波混合法正演模拟及稳定性分析[J].地球物理学报,2015,58(4):1290-1304.DU Qizhen,GUO Chengfeng,GONG Xufei.Hybrid PS/FD numerical simulation and stability analysis of pure P-wave propagation in VTI media[J].Chinese Journal of Geophysics,2015,58(4):1290-1304.
[5]
王璐琛,常旭,王一博.TTI介质的交错网格伪P波正演方法[J].地球物理学报,2016,59(3):1046-1058.WANG Luchen,CHANG Xu,WANG Yibo.Forward modeling of pseudo P waves in TTI medium using staggered grid[J].Chinese Journal of Geophysics,2016,59(3):1046-1058.
汪勇,段焱文,王婷,等.近似解析离散化方法的粘弹声波方程数值模拟及波场特征分析[J].石油物探,2017,56(3):362-372.WANG Yong,DUAN Yanwen,WANG Ting,et al.Numerical simulation and the wave field characteristics analysis of viscoelastic acoustic wave equation based on the nearly-analytic discrete method[J].Geophysical Prospecting for Petroleum,2017,56(3):362-372.
[8]
Druskin V,Guttel S,Knizhnerman L.Near-optimal perfectly matched layers for indefinite Helmholtz problems[J].Society for Industrial and Applied Mathematics Review,2015,58(1):90-116.
[9]
Rabinovich D,Dan G,Bielak J,et al.The double absorbing boundary method for a class of anisotropic elastic media[J].Computer Methods in Applied Mechanics & Engineering,2017,315:190-221.
[10]
Lee J H,Tassoulas J L.Absorbing boundary condition for scalar-wave propagation problems in infinite media based on a root-finding algorithm[J].Computer Methods in Applied Mechanics and Engineering,2018,330(1):207-219.
[11]
Jing H,Chen Y S,Yang D H,et al.Dispersion-relation preserving stereo-modeling method beyond Nyquist frequency for acoustic wave equation[J].Geophysics,2017,82(1):T1-T15.
[12]
Liu S L,Yang D H,Lan C,et al.Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations[J].Computer Physics Communications,2017,213:52-63.
[13]
汪勇,段焱文,王婷,等.优化近似解析离散化方法的二维弹性波波场分离模拟[J].石油地球物理勘探,2017,52(3):458-467.WANG Yong,DUAN Yanwen,WANG Ting,et al.Numerical simulation of elastic wave separation in 2D isotropic medium with the optimal nearly-analytic discretization[J].Oil Geophysical Prospecting,2017,52(3):458-467.
[14]
Du Q,Li B,Hou B.Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme[J].Applied Geophysics,2009,6(1):42-49.
[15]
Liu Y,Sen M K.A practical implicit finite-difference method:Examples from seismic modeling[J].Journal of Geophysics and Engineering,2009,6(3):231-249.
[16]
Liu Y,Sen M K.An implicit staggered-grid finite-difference method for seismic modeling[J].Geophy-sical Journal International,2009,179(1):459-474.
[17]
杨宽德,宋国杰,李静爽.Biot流动和喷射流动耦合作用下波传播的FCT紧致差分模拟[J].地球物理学报,2011,54(5):1348-1357.YANG Kuande,SONG Guojie,LI Jingshuang.FCT compact difference simulation of wave propagation based on the Biot and squirt-flow coupling interaction[J].Chinese Journal of Geophysics,2011,54(5):1348-1357.
[18]
Liao W Y.On the dispersion,stability and accuracy of a compact higher-order finite difference scheme for 3D acoustic wave equation[J].Journal of Computational and Applied Mathematics,2014,270:571-583.
[19]
Akbari R,Mokhtari R.A new compact finite difference method for solving the generalized long wave equation[J].Numerical Functional Analysis and Optimization,2014,35(2):133-152.
Córdova L, Rojas O, Otero B, et al. Compact finite difference modeling of 2-D acoustic wave propagation[J].Journal of Computational and Applied Mathema-tics,2015,295:83-91.
[22]
Wang Z K, Li J Y,Wang B F,et al.A new central compact finite difference scheme with high spectral resolution for acoustic wave equation[J].Journal of Computational Physics,2018,366(1):191-206.
[23]
Madariaga R.Dynamics of an expanding circular fault[J].Bulletin of the Seismological Society of America,1976,66(3):639-666.
[24]
董良国,马在田,曹景忠,等.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):411-419.DONG Liangguo,MA Zaitian,CAO Jingzhong,et al.The staggered-grid high-order difference method of first-order elastic equation[J].Chinese Journal of Geophysics,2000, 43(3):411-419.
[25]
Liu Y,Sen M K.A new time-space domain high-order finite-diffference method for the acoustic wave equation[J].Journal of Computational Physics,2009,228(1):8779-8806.
[26]
Yan H,Yang L,Li X Y.Optimal staggered-grid finite-difference schemes by combining Taylor-series expansion and sampling approximation for wave equation modeling[J].Journal of Computational Physics,2016,326(1):913-930.
[27]
Kim J W, Lee D J.Optimized compact finite difference schemes for computational acoustics[J].American Institute of Aeronautics and Astronautics Journal, 1996,34(5):887-893.
[28]
Liu Y,Sen M K.3D acoustic wave modeling with time-space domain dispersion relation based finite difference schemes and hybrid absorbing boundary conditions[J].Exploration Geophysics,2011,42(3):176-189.
[29]
Zhang J H,Yao Z X.Optimized finite-difference operator for broadband seismic wave modeling[J].Geophysics,2013,78(1):A13-A18.
[30]
Zhang J H,Yao Z X.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm[J].Journal of Computational Physics,2013,250(10):511-526.
[31]
Liu Y.Globally optimal finite-difference schemes based on least squares[J].Geophysics,2013,78(4):T113-T132.
[32]
Liu Y.Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modeling[J].Geophysical Journal International,2014,197(2):1033-1047.
[33]
Yu C H,Wang D,He Z,et al.An optimized dispersion-relation-preserving combined compact difference scheme to solve advection equations[J].Journal of Computational Physics,2015,300(1):92-115.
[34]
Ren Z,Liu Y.Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes[J].Geophysics,2015,80(1):T17-T40.
[35]
Yang L,Yan H,Liu H.Optimal staggered-grid finite-difference schemes based on the minimax approximation methos with the Remez algorithm[J].Geophy-sics,2017,82(1):T27-T42.
[36]
Yang L,Yan H Y,Liu H.An optimal implicit staggered-grid finite-difference scheme based on the mo-dified Taylor-series expansion with minimax approximation method for elastic modeling[J].Journal of Applied Geophysics,2017,138:161-171.
[37]
Levander A R.Fourth-order finite-difference P-SV seismograms[J].Geophysics,1988,53(11):1425-1436.
[38]
Nagarajan S,Lele S K,Ferziger J H.A robust high-order compact method for large eddy simulation[J].Journal of Computational Physics,2003,191(2):392-419.
[39]
Boersma B J.A staggered compact finite difference formulation for the compressible Navier-Stokes equations[J].Journal of Computational Physics,2005, 208(2):675-690.
[40]
Tam C K W,Webb J C.Dispersion-relation-preserving finite difference schemes for computational acoustics[J].Journal of Computational Physics,1993, 107(2):262-281.
