基于辛格式离散奇异褶积微分算子的弹性波场模拟
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摘要
本文发展了基于辛格式离散奇异褶积微分算子(SDSCD)的保结构方法模拟弹性波场,求解弹性波动方程时,引入辛差分格式进行时间离散,采用离散奇异褶积微分算子进行空间离散.相比于传统的伪谱方法,该方法提高了计算精度和稳定性.数值结果表明SDSCD方法可以有效地抑制数值频散,为解决大尺度、长时程地震波场模拟问题提供了合适的数值方法.
In this paper,we introduce a structure-preserving method based on symplectic discrete singular convolution differentiator(SDSCD) for simulating elastic wave fields.In the method presented for solving elastic wave equations,physical space is discretized by singular convolution differentiator,whereas a symplectic difference scheme is used for the time discretization.The computational accuracy and stability of this method have been greatly improved compared with traditional pseudo-spectral method.Numerical results suggest the SDSCD algorithm can suppress effectively numerical dispersion,and it is suitable for modeling the large-scale and long-term seismic wave propagation.
In this paper,we introduce a structure-preserving method based on symplectic discrete singular convolution differentiator(SDSCD) for simulating elastic wave fields.In the method presented for solving elastic wave equations,physical space is discretized by singular convolution differentiator,whereas a symplectic difference scheme is used for the time discretization.The computational accuracy and stability of this method have been greatly improved compared with traditional pseudo-spectral method.Numerical results suggest the SDSCD algorithm can suppress effectively numerical dispersion,and it is suitable for modeling the large-scale and long-term seismic wave propagation.
引文
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[4]钟万勰.应用力学的辛数学方法.北京:高等教育出版社,2006.Zhong W X.Symplectic Solution Methodology in AppliedMechanics(in Chinese).Beijing:Higher Education Press,2006.
[5]zdenvar T,McMechan G A.Causes and reduction ofnumerical artifacts in pseudo-spectral wavefield extrapolation.Geophys.J.Int.,1996,126(3):819-828.
[6]Li X F,Zhu T,Zhang M G,et al.Seismic scalar waveequation with variable coefficients modeling by a newconvolutional differentiator.Computer Physics Communications,2010,181(11):1850-1858.
[7]高亮,李幼铭,陈旭荣等.地震射线辛几何算法初探.地球物理学报,2000,43(3):402-410.Gao L,Li Y M,Chen X R,et al.An attempt to seismic raytracing with symplectic algorithm.Chinese J.Geophys.(inChinese),2000,43(3):402-410.
[8]李世雄.波动方程的高频近似与辛几何.北京:科学出版社,2001.Li S X.The High-Frequency Approximation of WaveEquations and Symplectic Algorithm(in Chinese).Beijing:Science Press,2001.
[9]陈景波,秦孟兆.辛几何算法在射线追踪中的应用.数值计算与计算机应用,2000,21(4):255-265.Chen J B,Qin M Z.Ray tracing by symplectic algorithm.Numerical Computation and Computer Application(inChinese),2000,21(4):255-265.
[10]Chen J B.Lax-Wendroff and Nystrm methods for seismicmodelling.Geophysical Prospecting,2009,57(6):931-941.
[11]Holberg O.Computational aspects of the choice of operatorand sampling interval for numerical differentiation in large-scale simulation of wave phenomena.Geophysical Prospecting,1987,35(6):629-655.
[12]Zhou B,Greenhalgh S A.Seismic scalar wave equationmodeling by a convolutional differentiator.Bull.Seism.Soc.Am.,1992,82(1):289-303.
[13]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63-69.Zhang Z J,Teng J W,Yang D H.The convolutionaldifferentiator method for numerical modeling of acoustic andelastic wave-field.Acta Seismologica Sinica(in Chinese),1996,18(1):63-69.
[14]戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报(地球科学版),2005,35(4):520-524.Dai Z Y,Sun J G,Zha X J.Seismic wave field modeling withconvolutional differentiator algorithm.Journal of JilinUniversity(Earth Science Edition)(in Chinese),2005,35(4):520-524.
[15]龙桂华,李小凡,张美根.基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用.地球物理学报,2009,52(4):1014-1024.Long G H,Li X F,Zhang M G.The application ofconvolutional differentiator in seismic modeling based onShannon singular kernel theory.Chinese J.Geophys.(inChinese),2009,52(4):1014-1024.
[16]李一琼,李小凡,朱童.基于辛格式奇异核褶积微分算子的地震标量波场模拟.地球物理学报,2011,54(7):1827-1834.Li Y Q,Li X F,Zhu T.The seismic scalar wave fieldmodeling by symplectic scheme and singular kernelconvolutional differentiator.Chinese J.Geophys.(in Chinese),2011,54(7):1827-1834.
[17]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-waveequation modeling by a multisymplectic discrete singularconvolution differentiator method.Bull.Seism.Soc.Am.,2011,101(4):1710-1718.
[18]Iwatsu R.Two new solutions to the third-order symplecticintegration method.Phys.Lett.A.,2009,373(34):3056-3060.
[19]Wei G W.Discrete singular convolution for the solution ofthe Fokker-Planck equations.J.Chem.Phys.,1999,110(18):8930-8942.
[20]黄志祥,沙威,吴先良等.高阶辛算法的稳定性与数值色散性分析.计算物理,2010,27(1):82-88.Huang Z X,Sha W,Wu X L,et al.Stability and numericaldispersion of high order symplectic schemes.Chinese Journalof Computational Physics(in Chinese),2010,27(1):82-88.
[21]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.地球物理学报,2000,43(6):856-864.Dong L G,Ma Z T,Cao J Z.A study on stability of thestaggered-grid high-order difference method of first-orderelastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6):856-864.
[22]Chen J B.Modeling the scalar wave equation with Nystrmmethods.Geophysics,2006,71(5):151-158.
[2]Fornberg B.The pseudospectral method:Comparisons withfinite differences for the elastic wave equation.Geophysics,1987,52(4):483-501.
[3]Smith W D.The application of finite element analysis to bodywave propagation problems.Geophys.J.Int.,1975,42(2):747-768.
[4]钟万勰.应用力学的辛数学方法.北京:高等教育出版社,2006.Zhong W X.Symplectic Solution Methodology in AppliedMechanics(in Chinese).Beijing:Higher Education Press,2006.
[5]zdenvar T,McMechan G A.Causes and reduction ofnumerical artifacts in pseudo-spectral wavefield extrapolation.Geophys.J.Int.,1996,126(3):819-828.
[6]Li X F,Zhu T,Zhang M G,et al.Seismic scalar waveequation with variable coefficients modeling by a newconvolutional differentiator.Computer Physics Communications,2010,181(11):1850-1858.
[7]高亮,李幼铭,陈旭荣等.地震射线辛几何算法初探.地球物理学报,2000,43(3):402-410.Gao L,Li Y M,Chen X R,et al.An attempt to seismic raytracing with symplectic algorithm.Chinese J.Geophys.(inChinese),2000,43(3):402-410.
[8]李世雄.波动方程的高频近似与辛几何.北京:科学出版社,2001.Li S X.The High-Frequency Approximation of WaveEquations and Symplectic Algorithm(in Chinese).Beijing:Science Press,2001.
[9]陈景波,秦孟兆.辛几何算法在射线追踪中的应用.数值计算与计算机应用,2000,21(4):255-265.Chen J B,Qin M Z.Ray tracing by symplectic algorithm.Numerical Computation and Computer Application(inChinese),2000,21(4):255-265.
[10]Chen J B.Lax-Wendroff and Nystrm methods for seismicmodelling.Geophysical Prospecting,2009,57(6):931-941.
[11]Holberg O.Computational aspects of the choice of operatorand sampling interval for numerical differentiation in large-scale simulation of wave phenomena.Geophysical Prospecting,1987,35(6):629-655.
[12]Zhou B,Greenhalgh S A.Seismic scalar wave equationmodeling by a convolutional differentiator.Bull.Seism.Soc.Am.,1992,82(1):289-303.
[13]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63-69.Zhang Z J,Teng J W,Yang D H.The convolutionaldifferentiator method for numerical modeling of acoustic andelastic wave-field.Acta Seismologica Sinica(in Chinese),1996,18(1):63-69.
[14]戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报(地球科学版),2005,35(4):520-524.Dai Z Y,Sun J G,Zha X J.Seismic wave field modeling withconvolutional differentiator algorithm.Journal of JilinUniversity(Earth Science Edition)(in Chinese),2005,35(4):520-524.
[15]龙桂华,李小凡,张美根.基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用.地球物理学报,2009,52(4):1014-1024.Long G H,Li X F,Zhang M G.The application ofconvolutional differentiator in seismic modeling based onShannon singular kernel theory.Chinese J.Geophys.(inChinese),2009,52(4):1014-1024.
[16]李一琼,李小凡,朱童.基于辛格式奇异核褶积微分算子的地震标量波场模拟.地球物理学报,2011,54(7):1827-1834.Li Y Q,Li X F,Zhu T.The seismic scalar wave fieldmodeling by symplectic scheme and singular kernelconvolutional differentiator.Chinese J.Geophys.(in Chinese),2011,54(7):1827-1834.
[17]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-waveequation modeling by a multisymplectic discrete singularconvolution differentiator method.Bull.Seism.Soc.Am.,2011,101(4):1710-1718.
[18]Iwatsu R.Two new solutions to the third-order symplecticintegration method.Phys.Lett.A.,2009,373(34):3056-3060.
[19]Wei G W.Discrete singular convolution for the solution ofthe Fokker-Planck equations.J.Chem.Phys.,1999,110(18):8930-8942.
[20]黄志祥,沙威,吴先良等.高阶辛算法的稳定性与数值色散性分析.计算物理,2010,27(1):82-88.Huang Z X,Sha W,Wu X L,et al.Stability and numericaldispersion of high order symplectic schemes.Chinese Journalof Computational Physics(in Chinese),2010,27(1):82-88.
[21]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.地球物理学报,2000,43(6):856-864.Dong L G,Ma Z T,Cao J Z.A study on stability of thestaggered-grid high-order difference method of first-orderelastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6):856-864.
[22]Chen J B.Modeling the scalar wave equation with Nystrmmethods.Geophysics,2006,71(5):151-158.
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