二维地震波场的五点八阶超紧致有限差分数值模拟
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摘要
首先将迎风机制引入五点八阶超紧致有限差分(CCD8)格式,得到五点七阶迎风超紧致(UCCD7)格式,并对两种格式进行数值频散分析和精度分析;其次建立了二阶声波方程的位移场时间四阶离散格式,将五点CCD8格式和五点UCCD7格式分别应用于位移场空间导数的求取,并分析这两种格式的稳定性条件;最后基于PML边界条件,将上述两种格式分别应用于声波方程的均匀介质、水平层状介质及Marmousi模型的数值模拟和波场特征分析及对比。研究结果表明:相较于普通紧致差分,五点CCD8格式具有小截断误差、高模拟精度、低数值频散、高稳定性、所需网格点数少的优点;引入迎风机制后,声波方程的五点UCCD7格式稳定性得到进一步提高;模型试算的结果验证了五点CCD8格式适用于复杂介质的数值模拟,模拟精度和计算效率都高。
In this study the fifth-point eighth-order combined compact difference scheme(CCD8) and fifth-point seventh-order upwind combined compact difference scheme(UCCD7) were used in a numerical simulation based on acoustic and elastic wave equations.Based on Taylor series expansion and acoustic wave equation,a fourth-order discrete scheme for the displacement field was established.The CCD8 and UCCD7 were used to calculate the spatial derivative of the displacement field,and the accuracy,dispersion,and stability of the two schemes were analyzed.Finally,using the PML boundary conditions,the two formats were tested on uniform medium,horizontal layered medium,and Marmousi model.The results showed that compared with normal compact difference,CCD8 has smaller truncation error,higher simulation precision,lower numerical dispersion,higher stability,and requires fewer grid points;the introduction of windward mechanism could improve the stability of CCD7.The numerical simulation results verified that the CCD8 scheme and UCCD7 scheme are practical and effective.
In this study the fifth-point eighth-order combined compact difference scheme(CCD8) and fifth-point seventh-order upwind combined compact difference scheme(UCCD7) were used in a numerical simulation based on acoustic and elastic wave equations.Based on Taylor series expansion and acoustic wave equation,a fourth-order discrete scheme for the displacement field was established.The CCD8 and UCCD7 were used to calculate the spatial derivative of the displacement field,and the accuracy,dispersion,and stability of the two schemes were analyzed.Finally,using the PML boundary conditions,the two formats were tested on uniform medium,horizontal layered medium,and Marmousi model.The results showed that compared with normal compact difference,CCD8 has smaller truncation error,higher simulation precision,lower numerical dispersion,higher stability,and requires fewer grid points;the introduction of windward mechanism could improve the stability of CCD7.The numerical simulation results verified that the CCD8 scheme and UCCD7 scheme are practical and effective.
引文
[1] 孙林洁,刘春成,张世鑫.PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J].石油物探,2015,54(6):652-664SUN L J,LIU C C,ZHANG S X.A variable grid finite-difference scheme with PML boundary condition in elastic wave of porous media[J].Geophysical Prospecting for Petroleum,2015,54(6):652-664
[2] KOSLOFF D,BAYSAL E.Forward modeling by a Fourier method[J].Geophysics,1982,47(10):1402-1412
[3] 郭宏伟,王尚旭,孙文博.基于非均质体的波动方程有限元正演模拟[J].石油物探,2012,51(4):319-326GUO H W,WANG S X,SUN W B.Wave equation modeling by finite-element method for heterogeneous body[J].Geophysical Prospecting for Petroleum,2012,51(4):319-326
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[5] KOMATISSCH D,BARNES C,TROMP J.Simulation of anisotropic wave propagation based upon a spectral element method[J].Geophysics,2000,65(4):1251-1260
[6] 王颖,周辉,盛善波.贴体坐标系二维声波方程SBP有限差分法的稳定性分析[J].石油物探,2016,55(1):33-40WANG Y,ZHOU H,SHENG S B.Stability analysis of SBP finite difference scheme for two-dimensional acoustic wave equation in boundary-conforming grids[J].Geophysical Prospecting for Petroleum,2016,55(1):33-40
[7] 黄金强,李振春,黄建平,等.TTI介质Lebedev网格高阶有限差分正演模拟及波型分离[J].石油地球物理勘探,2017,52(5):915-927HUANG J Q,LI Z C,HUANG J P,et al.Lebedev grid high-order finite-difference modeling and elastic wave-mode separation for TTI media[J].Oil Geophysical Prospecting,2017,52(5):915-927
[8] ADAMS N A,SHARIFF K.A high-resolution hybrid compact-ENO scheme for Shock-Turbulence interaction problems[J].Journal of Computational Physics,1996,127(1):27-51
[9] 王芳芳.紧致差分格式的理论与分析[D].沈阳:东北大学,2010 WANG F F.Theory and analysis of compact difference scheme[D].Shenyang:Northeastern University,2010
[10] 柳占新,黄其柏,胡溧,等.计算气动声学中的高精度紧致差分格式研究[J].航空动力学报,2009,24(1):83-90LIU Z X,HUANG Q B,HU L,et al.Study on the high-accuracy compact-finite-difference schemes for computational aeroacoustics[J].Journal of Aerospace Power,2009,24(1): 83-90
[11] DENNIS S C,HUNDSON J D.Compact h4 finite-difference approximations to operators of Navier-Stokes type[J].Journal of Computational Physics,1989,85(2):390-416
[12] LELE S K.Compact finite difference scheme with spectral-like resolution[J].Journal of Computational Physics,1992,103(1):16-42
[13] CHU P C,FAN C W.A three-point combined compact difference scheme[J].Journal of Computational Physics,1998,140(2):370-399
[14] 林东,詹杰民.浅水方程超紧致差分格式[J].计算力学学报,2008,25(6):791-796 LIN D,ZHAN J M.Combined super compact finite difference scheme and application to simulation of shallow water equations[J].Chinese Journal of Computational Mechanics,2008,25(6):791-796
[15] 傅德薰.计算空气动力学[M].北京:中国宇航出版社,1994:183-189FU D X.Computational aerodynamics[M].Beijing:China Aerospace Publishing House,1994:183-189
[16] FU D X,MA Y W.A high order accuracy difference scheme for complex flow fields[J].Journal of Computational Physics,1997,134(1):1-15
[17] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法[J].清华大学学报(自然科学版),2002,42(8):1128-1131WANG S Q,YANG D H,YANG K D.Compact finite difference scheme for elastic equations[J].Journal of Tsinghua University(Science and Technology),2002,42(8):1128-1131
[18] KRISHNAN M.A family of high order finite difference schemes with good spectral resolution[J].Journal of Computational Physics,1998,145(2):332-358
[19] 贾伟.几种高精度紧致差分格式的构造与应用[D].兰州:西北师范大学,2015JIA W.The construction and application of several high accuracy compact difference schemes[D].Lanzhou:Northwest Normal University,2015
[20] EHATERINARIS J A.Implicit high-resolution compact schemes for gas dynamics and aeroacoustics[J].Journal of Computational Physics,1999,156(2):272-299
[21] 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):411-419DONG L G,MA Z T,CAO J Z,et al.The staggered-grid high-order difference method of first-order elastic equation[J].Chinese Journal of Geophysics,2000,43(3):411-419
[22] BLAYO E.Compact finite difference schemes for ocean models[J].Journal of Computational Physics,2000,164(2):241-257
[23] SENGUPTA T K,GANERIWAL G,DE S.Analysis of central and upwind compact schemes[J].Journal of Computational Physics,2003,192(2):677-694
[24] 梁文全,王彦飞,杨长春.声波方程数值模拟矩形网格有限差分系数确定法[J].石油地球物理勘探,2017,52(1):56-62LIANG W Q,WANG Y F,YANG C C.Acoustic wave equation modeling with rectangle grid finite difference operator and its linear time space domain solution[J].Oil Geophysical Prospecting,2017,52(1):56-62
[25] 吴国忱,王华忠.波场模拟中的数值频散分析与校正策略[J].地球物理学进展,2005,20(1):58-65WU G C,WANG H Z.Analysis of numerical dispersion in wave-field simulation[J].Progress in Geophysics,2005,20(1):58-65
[26] NIHEI T,ISHII K.A fast solver of the shallow water equations on a sphere using a combined compact difference scheme[J].Journal of Computational Physics,2003,187(2):639-659
[27] ROBERT M C,ZHAO J C.Compact finite difference method for integro-differential equations[J].Applied Mathematics and Computation,2006,177(1):271-288
[28] 张文生.科学计算中的偏微分方程有限差分法[M].北京:高等教育出版社,2006:49-53ZHANG W S.Finite difference methods for partial differential equations in science computation[M].Beijing:Higher Education Press,2006:49-53
[29] 柯璇,石颖,宋利伟,等.基于褶积完全匹配吸收边界的声波方程数值模拟[J].石油物探,2017,56(5):637-643KE X,SHI Y,SONG L W,et al.Numerical modeling of acoustic wave equations based on convolutional perfectly matched layer absorbing boundary condition[J].Geophysical Prospecting for Petroleum,2017,56(5):637-643
[30] 张衡,刘洪,李博,等.VTI介质声波方程非分裂式PML吸收边界条件研究[J].石油物探,2016,55(6):781-792ZHANG H,LIU H,LI B,et al.The research on unsplit PML absorbing boundary conditions of acoustic for VTI media[J].Geophysical Prospecting for Petroleum,2016,55(6):781-792
[31] 吴国忱,梁锴.VTI介质准P波频率空间域组合边界条件研究[J].石油物探,2005,44(4):301-307WU G C,LIANG K.Combined boundary conditions of quasi-P wave within ferquency-space domain in VTI media[J].Geophysical Prospecting for Petroleum,2005,44(4):301-307
[32] 王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34WANG S D.Absorbing boundary condition for acoustic wave equation by perfectly matched layer[J].Oil Geophysical Prospecting,2003,38(1):31-34
[2] KOSLOFF D,BAYSAL E.Forward modeling by a Fourier method[J].Geophysics,1982,47(10):1402-1412
[3] 郭宏伟,王尚旭,孙文博.基于非均质体的波动方程有限元正演模拟[J].石油物探,2012,51(4):319-326GUO H W,WANG S X,SUN W B.Wave equation modeling by finite-element method for heterogeneous body[J].Geophysical Prospecting for Petroleum,2012,51(4):319-326
[4] BOUCHON M,COUTANT O.Calculation of synthetic seismograms in a laterally varying medium by the boundary element-discrete wavenumber method[J].Bulletin of the Seismological Society of America,1994,84(6):1869-1881
[5] KOMATISSCH D,BARNES C,TROMP J.Simulation of anisotropic wave propagation based upon a spectral element method[J].Geophysics,2000,65(4):1251-1260
[6] 王颖,周辉,盛善波.贴体坐标系二维声波方程SBP有限差分法的稳定性分析[J].石油物探,2016,55(1):33-40WANG Y,ZHOU H,SHENG S B.Stability analysis of SBP finite difference scheme for two-dimensional acoustic wave equation in boundary-conforming grids[J].Geophysical Prospecting for Petroleum,2016,55(1):33-40
[7] 黄金强,李振春,黄建平,等.TTI介质Lebedev网格高阶有限差分正演模拟及波型分离[J].石油地球物理勘探,2017,52(5):915-927HUANG J Q,LI Z C,HUANG J P,et al.Lebedev grid high-order finite-difference modeling and elastic wave-mode separation for TTI media[J].Oil Geophysical Prospecting,2017,52(5):915-927
[8] ADAMS N A,SHARIFF K.A high-resolution hybrid compact-ENO scheme for Shock-Turbulence interaction problems[J].Journal of Computational Physics,1996,127(1):27-51
[9] 王芳芳.紧致差分格式的理论与分析[D].沈阳:东北大学,2010 WANG F F.Theory and analysis of compact difference scheme[D].Shenyang:Northeastern University,2010
[10] 柳占新,黄其柏,胡溧,等.计算气动声学中的高精度紧致差分格式研究[J].航空动力学报,2009,24(1):83-90LIU Z X,HUANG Q B,HU L,et al.Study on the high-accuracy compact-finite-difference schemes for computational aeroacoustics[J].Journal of Aerospace Power,2009,24(1): 83-90
[11] DENNIS S C,HUNDSON J D.Compact h4 finite-difference approximations to operators of Navier-Stokes type[J].Journal of Computational Physics,1989,85(2):390-416
[12] LELE S K.Compact finite difference scheme with spectral-like resolution[J].Journal of Computational Physics,1992,103(1):16-42
[13] CHU P C,FAN C W.A three-point combined compact difference scheme[J].Journal of Computational Physics,1998,140(2):370-399
[14] 林东,詹杰民.浅水方程超紧致差分格式[J].计算力学学报,2008,25(6):791-796 LIN D,ZHAN J M.Combined super compact finite difference scheme and application to simulation of shallow water equations[J].Chinese Journal of Computational Mechanics,2008,25(6):791-796
[15] 傅德薰.计算空气动力学[M].北京:中国宇航出版社,1994:183-189FU D X.Computational aerodynamics[M].Beijing:China Aerospace Publishing House,1994:183-189
[16] FU D X,MA Y W.A high order accuracy difference scheme for complex flow fields[J].Journal of Computational Physics,1997,134(1):1-15
[17] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法[J].清华大学学报(自然科学版),2002,42(8):1128-1131WANG S Q,YANG D H,YANG K D.Compact finite difference scheme for elastic equations[J].Journal of Tsinghua University(Science and Technology),2002,42(8):1128-1131
[18] KRISHNAN M.A family of high order finite difference schemes with good spectral resolution[J].Journal of Computational Physics,1998,145(2):332-358
[19] 贾伟.几种高精度紧致差分格式的构造与应用[D].兰州:西北师范大学,2015JIA W.The construction and application of several high accuracy compact difference schemes[D].Lanzhou:Northwest Normal University,2015
[20] EHATERINARIS J A.Implicit high-resolution compact schemes for gas dynamics and aeroacoustics[J].Journal of Computational Physics,1999,156(2):272-299
[21] 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):411-419DONG L G,MA Z T,CAO J Z,et al.The staggered-grid high-order difference method of first-order elastic equation[J].Chinese Journal of Geophysics,2000,43(3):411-419
[22] BLAYO E.Compact finite difference schemes for ocean models[J].Journal of Computational Physics,2000,164(2):241-257
[23] SENGUPTA T K,GANERIWAL G,DE S.Analysis of central and upwind compact schemes[J].Journal of Computational Physics,2003,192(2):677-694
[24] 梁文全,王彦飞,杨长春.声波方程数值模拟矩形网格有限差分系数确定法[J].石油地球物理勘探,2017,52(1):56-62LIANG W Q,WANG Y F,YANG C C.Acoustic wave equation modeling with rectangle grid finite difference operator and its linear time space domain solution[J].Oil Geophysical Prospecting,2017,52(1):56-62
[25] 吴国忱,王华忠.波场模拟中的数值频散分析与校正策略[J].地球物理学进展,2005,20(1):58-65WU G C,WANG H Z.Analysis of numerical dispersion in wave-field simulation[J].Progress in Geophysics,2005,20(1):58-65
[26] NIHEI T,ISHII K.A fast solver of the shallow water equations on a sphere using a combined compact difference scheme[J].Journal of Computational Physics,2003,187(2):639-659
[27] ROBERT M C,ZHAO J C.Compact finite difference method for integro-differential equations[J].Applied Mathematics and Computation,2006,177(1):271-288
[28] 张文生.科学计算中的偏微分方程有限差分法[M].北京:高等教育出版社,2006:49-53ZHANG W S.Finite difference methods for partial differential equations in science computation[M].Beijing:Higher Education Press,2006:49-53
[29] 柯璇,石颖,宋利伟,等.基于褶积完全匹配吸收边界的声波方程数值模拟[J].石油物探,2017,56(5):637-643KE X,SHI Y,SONG L W,et al.Numerical modeling of acoustic wave equations based on convolutional perfectly matched layer absorbing boundary condition[J].Geophysical Prospecting for Petroleum,2017,56(5):637-643
[30] 张衡,刘洪,李博,等.VTI介质声波方程非分裂式PML吸收边界条件研究[J].石油物探,2016,55(6):781-792ZHANG H,LIU H,LI B,et al.The research on unsplit PML absorbing boundary conditions of acoustic for VTI media[J].Geophysical Prospecting for Petroleum,2016,55(6):781-792
[31] 吴国忱,梁锴.VTI介质准P波频率空间域组合边界条件研究[J].石油物探,2005,44(4):301-307WU G C,LIANG K.Combined boundary conditions of quasi-P wave within ferquency-space domain in VTI media[J].Geophysical Prospecting for Petroleum,2005,44(4):301-307
[32] 王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34WANG S D.Absorbing boundary condition for acoustic wave equation by perfectly matched layer[J].Oil Geophysical Prospecting,2003,38(1):31-34