弹性波和声波的时域仿真方法研究
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摘要
时域有限差分(FDTD)法是求解电磁波、弹性波、声波等波动问题的一种数值计算方法,它把带时间变量的波动微分方程转化为差分方程来求解波场各分量,得到时间系列的场分量的空间分布。自FDTD法创建以来,它广泛应用于电磁波、弹性波和声波的传播过程,以及与目标相互作用过程的研究中。
超声检测是一种有效的医疗诊断技术,其物理基础是超声波在生物组织中的传播规律,也即利用各种组织声学特性的差异来区分不同组织,特别是区分正常组织和病变组织。超声传播规律的研究对诊断是十分重要的,其方法是先对生物组织进行复杂的建模,然后建立这个模型的波动方程,根据已知声源,求解声波在模型中的传播规律,从而得到声波在该生物组织中传播规律的近似。时域有限差分法对复杂模型的适用性,预示着这是研究超声传播规律的有效技术。本文进行时域有限差分法研究的目的就是解决医学超声探测的动态计算机仿真问题。
生物组织的不同建模要求涉及到弹性波和声波问题。本文研究了弹性波和声波领域的时域有限差分原理,包括二维和三维问题的FDTD计算; 对FDTD算法的稳定性条件和数值色散做了简单的分析; 讨论并选取了激励源的合适形式,以及激励源的设置方式。
吸收边界条件是时域有限差分法的一个重要问题,FDTD模拟的是有限区域的波场计算,因而必须在截断边界处设置特殊边界来吸收外向行波。本文采用了分解场分量的完全匹配层(PML)吸收边界条件来吸收外向行波,并对其进行改进,应用在弹性波和声波领域。
最后,本文对弹性波和声波领域的FDTD算法的稳定性和PML吸收边界条件的吸收效果进行了检验。结果表明,本文编写的FDTD程序运行稳定,且PML边界吸收效果良好。利用该程序对弹性波和声波与目标相互作用问题进行了二维和三维的FDTD计算,也对分层异类媒质中的波动传播进行了FDTD模拟。
The finite-difference time-domain method (FDTD) is one of the most powerful numerical methods for the forward modeling of the electromagnetic wave, elastic wave and acoustic wave propagation and scattering. Through transforming the partial-differential equations to the finite-difference equations, the wave field weight is obtained in the whole space. The FDTD method is a full wave technique that has been widely used to model wave propagation process,and interaction with certain or random object in electromagnetics, acoustics, and elastodynamics.
Ultrasonic exploration is an efficacious medical diagnosis technique. The ultrasonic propagation rule in the biology tissue is the physics foundation of the ultrasonic diagnosis. Based on the difference of acoustics characteristics in different biology tissues, it distinguishes the dissimilar tissues, especially the natural tissue from the pathological tissue. The investigation of the ultrasonic propagation rule is important for the medical treatment. First, the method is to develop a complex model of the biology tissue, then to establish the wave equations of this model, and to seek to the acoustic propagation rule in the model base on the known source. Finally the approximate acoustic propagation rule in the biology tissue is found. The FDTD method is an effective technique for the research about ultrasonic propagation rule, because the FDTD is especially appropriate for complex models. The purpose of research on FDTD is to obtain the dynamic computer model of the medical ultrasonic exploration.
The various modeling request of the biology tissue come down to the elastic wave and acoustic wave questions. In this thesis, the FDTD theory of the elastic and acoustic wave is mainly discussed, including two dimensional (2D) and three dimensional (3D) FDTD simulations. Then the stability condition of the FDTD method and numerical dispersive are analyzed,and the form and the sort of the power source, and the setting of the source are discussed and adopted.
Absorbing boundary condition is an important problem of the FDTD method. Because FDTD simulates the wave field in the finite area, we must mount an especial boundary to absorb the outward travelling wave in the truncation boundary. A perfectly matched layer (PML) absorbing boundary condition is used to absorb the
outward travelling wave, which is improved and applied in the elastic and acoustic wave simulation. In the end, the thesis verifies the stability of the elastic and acoustic wave’s FDTD method and the absorbing effect of the PML absorbing boundary condition. The result proves that the FDTD programs by the writer run stably and the PML boundary can absorb the outward travelling wave well. Moreover, we simulate the interaction of the elastic wave, acoustic wave and the object through 2D/3D FDTD programs, as well as the propagation of the elastic wave in the heterogeneous layered media.
超声检测是一种有效的医疗诊断技术,其物理基础是超声波在生物组织中的传播规律,也即利用各种组织声学特性的差异来区分不同组织,特别是区分正常组织和病变组织。超声传播规律的研究对诊断是十分重要的,其方法是先对生物组织进行复杂的建模,然后建立这个模型的波动方程,根据已知声源,求解声波在模型中的传播规律,从而得到声波在该生物组织中传播规律的近似。时域有限差分法对复杂模型的适用性,预示着这是研究超声传播规律的有效技术。本文进行时域有限差分法研究的目的就是解决医学超声探测的动态计算机仿真问题。
生物组织的不同建模要求涉及到弹性波和声波问题。本文研究了弹性波和声波领域的时域有限差分原理,包括二维和三维问题的FDTD计算; 对FDTD算法的稳定性条件和数值色散做了简单的分析; 讨论并选取了激励源的合适形式,以及激励源的设置方式。
吸收边界条件是时域有限差分法的一个重要问题,FDTD模拟的是有限区域的波场计算,因而必须在截断边界处设置特殊边界来吸收外向行波。本文采用了分解场分量的完全匹配层(PML)吸收边界条件来吸收外向行波,并对其进行改进,应用在弹性波和声波领域。
最后,本文对弹性波和声波领域的FDTD算法的稳定性和PML吸收边界条件的吸收效果进行了检验。结果表明,本文编写的FDTD程序运行稳定,且PML边界吸收效果良好。利用该程序对弹性波和声波与目标相互作用问题进行了二维和三维的FDTD计算,也对分层异类媒质中的波动传播进行了FDTD模拟。
The finite-difference time-domain method (FDTD) is one of the most powerful numerical methods for the forward modeling of the electromagnetic wave, elastic wave and acoustic wave propagation and scattering. Through transforming the partial-differential equations to the finite-difference equations, the wave field weight is obtained in the whole space. The FDTD method is a full wave technique that has been widely used to model wave propagation process,and interaction with certain or random object in electromagnetics, acoustics, and elastodynamics.
Ultrasonic exploration is an efficacious medical diagnosis technique. The ultrasonic propagation rule in the biology tissue is the physics foundation of the ultrasonic diagnosis. Based on the difference of acoustics characteristics in different biology tissues, it distinguishes the dissimilar tissues, especially the natural tissue from the pathological tissue. The investigation of the ultrasonic propagation rule is important for the medical treatment. First, the method is to develop a complex model of the biology tissue, then to establish the wave equations of this model, and to seek to the acoustic propagation rule in the model base on the known source. Finally the approximate acoustic propagation rule in the biology tissue is found. The FDTD method is an effective technique for the research about ultrasonic propagation rule, because the FDTD is especially appropriate for complex models. The purpose of research on FDTD is to obtain the dynamic computer model of the medical ultrasonic exploration.
The various modeling request of the biology tissue come down to the elastic wave and acoustic wave questions. In this thesis, the FDTD theory of the elastic and acoustic wave is mainly discussed, including two dimensional (2D) and three dimensional (3D) FDTD simulations. Then the stability condition of the FDTD method and numerical dispersive are analyzed,and the form and the sort of the power source, and the setting of the source are discussed and adopted.
Absorbing boundary condition is an important problem of the FDTD method. Because FDTD simulates the wave field in the finite area, we must mount an especial boundary to absorb the outward travelling wave in the truncation boundary. A perfectly matched layer (PML) absorbing boundary condition is used to absorb the
outward travelling wave, which is improved and applied in the elastic and acoustic wave simulation. In the end, the thesis verifies the stability of the elastic and acoustic wave’s FDTD method and the absorbing effect of the PML absorbing boundary condition. The result proves that the FDTD programs by the writer run stably and the PML boundary can absorb the outward travelling wave well. Moreover, we simulate the interaction of the elastic wave, acoustic wave and the object through 2D/3D FDTD programs, as well as the propagation of the elastic wave in the heterogeneous layered media.
引文
[1] Yee K. S., Numerical solution of initial boundary value problem involving Maxwell equation in isotropic media. IEEE Trans. Antennas Propagat., May 1996, AP-14(3):302～307
[2] Shlager K L and Schneider J B, A selective survey of the finite-difference time-domain literature. IEEE Antennas Propagat. Magazine, Aug.1995, 37(4):39～57
[3] 葛德彪,闫玉波,电磁波时域有限差分方法,西安:西安电子科技大学出版社,2002.4
[4] Taflove A. and M. E. Brodwin, Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell' equations. IEEE Trans. Microwave Theory and Techniques, 1975, 23:623～630
[5] Taflove A. and M. E. Brodwin, Computation of the electro-magnetic scattering problems fields and induced temperatures within a model of the microwave-irradiated human eye. IEEE Trans. Microwave Theory and Techniques, 1975, 23:888～986
[6] Taylor C D, Lam D H and Shumpert T H, EM pulse scattering in time varying inhomogeneous media. IEEE Trans. Antennas Propagat., Sept.1969, AP-17(5):585～589
[7] Mur G, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations. IEEE Trans. Electramagn. Comput., Nov.1981, EMC-23(4):377～382
[8] Berenger J P, A perfectly matched layer for the absorption of the electromagnetic waves. J. Comput. Phys., Oct.1994, 114(2):185～200
[9] Berenger J P, Three-dimensional perfectly matched layer for the absorption of the electromagnetic waves. J. Comput. Phys., Sept.1996, 127(2):363～379
[10]Berenger J P, Perfectly matched layer for the FDTD solution of wave-structure interaction problem. J. Comput. Phys., Sept.1996, AP-44(2):110～117
[11] Sacks Z S, Kingsland D M, Lee D M and Lee J F, A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propagat., Dec.1995, AP-43(12):1460～1463
[12] Gedney S D, An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices. IEEE Trans. Antennas Propagat., Dec.1996, AP-44(12):1630～1639
[13] Clayton R and Egquist, Absorbing boundary Conditions for Acoustic and Elastic Wave Equations. Bull. Seism. Soc. Am., 1977, 67:1529～1540
[14] Clayton R and Egquist, Absorbing boundary Conditions for Wave-Equation Migration. Geophysics, 1980, 45:895～904
[15] Givoli D, Review: Non-reflecting boundary conditions. J. Comput. Phys., 1991, 94:1～29
[16] Higdon R L, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math. Comp., 1986, 47:437～459
[17] Higdon R L, Numerical Absorbing Boundary Conditions for the Wave Equation. Math. Comp., 1987, 49:65～90
[18] Higdon R L, Absorbing Boundary Conditions for Elastic Waves. Geophysics, 1991, 56(2):231～241
[19] Higdon R L, Absorbing Boundary Conditions for Acoustic and Elastic Waves in Stratified Media. J. Comp. Phys., 1992, 101:386～418
[20] Chew W. C. and Liu Q. H., Perfectly Matched Layers for Elastodynamics: A New Absorbing Boundary Condition. J. Comp. Acoust., Dec.1996, 4(4):72～79
[21] Yuan Xiaojuen, Borup D., Wiskin J.W., Berggren M., Eidens R., Johnson S.A., Formulation and validation of Berenger’s PML absorbing boundary for the FDTD simulation of acoustic scattering. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, July 1997, 44(4):816～822
[22] Quan Qi and Thomas L. Geers, Evaluation of the Perfectly Matched Layer for Computational Acoustics. J. Comp. Phys., 1998, 139:166～183
[23] Francis Collino and Chrysoula Tsogka, Application of the perfectly matched absorbing layer model to the linear elastodynamic problemin anisotropic heterogeneous media. Geophysics, January-February 2001, 66(1):294～307
[24] Mur G, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations, IEEE Tras. Electromagnetic Compatibility, 1981, 23:377～382
[25] Umashankar K. R. and A. Taflove, A novel method to analyze electromagnetic scattering of complex objects. IEEE Trans. Electromagnetic Compatibility, 1982, 24:397～405
[26] Schneider J. B., Wagner C. L., Ramahi O. M., Implementation of transparent sources in FDTD simulations. IEEE Trans. Antennas Propagat., Aug.1998, 46(8):1159～1168
[27] Schneider J. B., Wagner C. L. and Broschat S. L., Implementation of Transparent Sources Embedded in Acoustic Finite-Difference Time-Domain Grids. Journal of the Acoustical Society of America, 1998, 103(1):136～142
[28]王长清,祝西里编著,电磁场计算中的时域有限差分法,北京:北京大学出版社,1993.5
[29] C T, A Finite-Difference Model for Elastic Waves in the Ground. Diploma thesis, Univ. Braunschweig, Braunschweig, Germany, 1999 Schr& o&der
[30] Virieux J, SH-wave propagation in heterogenous media: Velocity-stress finite-difference method. Geophys., Nov.1984, 49: 1933～1957
[31] Virieux J., P-SV wave propagation in heterogenous media: Velociy-stress finite-difference method. Geophys., Apr.1986, 51:889～901
[32] Botteldooren D, Finite-difference time-domain simulation of low-frequency room acoustic problems. J. Acoust. Soc. Am., 1995, 98(6):3302～3308.
[33] Botteldooren D, Acoustical finite-difference time-domain simulation in a quasiCartesian grid. J. Acoust. Soc. Am., 1994, 95(5): 2313～2319.
[34] Maloney J. G. and Cummings K. E., Adaptation of FDTD techniques to acoustic modeling. in 11th Annual Review of Progress in Applied Computational Electronagnetics, Mar.1995, Vol.2, Monterey, CA: 724～731.
[35] 喻志远,时域有限差分计算中源的研究,电子科技大学学报,1998,27(1):43～46
[36] Yu Z., A Bandpass Source technique for the FDTD Analysis of Waveguide Discontinuity. Microw. Opt. Technol. Lett., 1998, 17(2):132～135
[37] 赵安平,于荣金,刘立杰,一种用于时域有限差分分析波导和微带电路的激励源技术,电子学报,1998,26(2):57～64
[38] Alterman Z. S. and F. C. Jr. Karal, Propagation of Elastic Waves in Layered Media by Finite-Difference Methods. Bull. Seism. Soc. Am., 1968, 58:367～398
[39] Vidale J E and Helmberger D V, Elastic Finite-Difference Modeling of the 1971 San Fernando, California Earthquake. Bull. Seism. Soc. Am., 1988, 78:122～141
[40] Helmberger D V and Vidale J E, Modeling Strong Motions Produced by Earthquakes with Two-Dimesional Numerical Codes. Bull. Seism. Soc. Am., 1988, 78:109～121
[41] Boore D M, Finite-Difference Methods for Seismic Wave Propagation in Heterogeneous Materials. In Vol. 11, Methods in Computational Physics (ed. Bolt, B. A.), Academic Press, New York, 1972
[42] Schr?der C T, Scott W R, On the Stability of the FDTD Algorithm for Elastic Media at a Material Interface [J]. IEEE Transactions on Geoscience and Remote Sensing, 2002, 40(2):474～481
[43] Jurgens T. G., A. Taflove, K. Umashankar, and T. G. Moore, Finite-difference time-domain modeling of curved surfaces, IEEE Trans. Antennas Propagat., Apr.1992, 40:357～366
[44] 廖振鹏著,工程波动理论导论,北京:科学技术出版社,2002.3
[45] Trefethen L N and Halpern L, Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions. Math. Comput., 1986, 47:421～435
[46] Halpern L and Trefethen L N, Wide-angle One-Way Wave Equations. J. Acoust. Soc. Am., 1988, 84:1397～1404
[47] Kolakowski H, Remarks on Perfectly Absorbing Boundary Conditions. Math. Meth. in the Appl. Sci., 1985, 7:486～492
[48] Kolakowski H, Absorbing Boundary Conditions for the Linearized Shallow Water Equations. Math. Meth. in the Appl. Sci., 1986, 8:41～49
[49] Halpern L and Rauch J, Error Analysis for Absorbing Boundary Conditions. Numer. Math., 1987, 51:459～467
[50] Higdon R L, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math. Comp., 1986, 47:437～459
[51] Keys R G, Absorbing Boundary Conditions for Acoustic Media. Geophys, 1985, 50:892～902
[52] Higdon R L, Numerical Absorbing Boundary Conditions for the Wave Equation. Math. Comp., 1987, 49:65～90
[53] Higdon R L, Absorbing Boundary Conditions for Elastic Waves. Geophysics, 1991, 56(2):231～241
[2] Shlager K L and Schneider J B, A selective survey of the finite-difference time-domain literature. IEEE Antennas Propagat. Magazine, Aug.1995, 37(4):39～57
[3] 葛德彪,闫玉波,电磁波时域有限差分方法,西安:西安电子科技大学出版社,2002.4
[4] Taflove A. and M. E. Brodwin, Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell' equations. IEEE Trans. Microwave Theory and Techniques, 1975, 23:623～630
[5] Taflove A. and M. E. Brodwin, Computation of the electro-magnetic scattering problems fields and induced temperatures within a model of the microwave-irradiated human eye. IEEE Trans. Microwave Theory and Techniques, 1975, 23:888～986
[6] Taylor C D, Lam D H and Shumpert T H, EM pulse scattering in time varying inhomogeneous media. IEEE Trans. Antennas Propagat., Sept.1969, AP-17(5):585～589
[7] Mur G, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations. IEEE Trans. Electramagn. Comput., Nov.1981, EMC-23(4):377～382
[8] Berenger J P, A perfectly matched layer for the absorption of the electromagnetic waves. J. Comput. Phys., Oct.1994, 114(2):185～200
[9] Berenger J P, Three-dimensional perfectly matched layer for the absorption of the electromagnetic waves. J. Comput. Phys., Sept.1996, 127(2):363～379
[10]Berenger J P, Perfectly matched layer for the FDTD solution of wave-structure interaction problem. J. Comput. Phys., Sept.1996, AP-44(2):110～117
[11] Sacks Z S, Kingsland D M, Lee D M and Lee J F, A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propagat., Dec.1995, AP-43(12):1460～1463
[12] Gedney S D, An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices. IEEE Trans. Antennas Propagat., Dec.1996, AP-44(12):1630～1639
[13] Clayton R and Egquist, Absorbing boundary Conditions for Acoustic and Elastic Wave Equations. Bull. Seism. Soc. Am., 1977, 67:1529～1540
[14] Clayton R and Egquist, Absorbing boundary Conditions for Wave-Equation Migration. Geophysics, 1980, 45:895～904
[15] Givoli D, Review: Non-reflecting boundary conditions. J. Comput. Phys., 1991, 94:1～29
[16] Higdon R L, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math. Comp., 1986, 47:437～459
[17] Higdon R L, Numerical Absorbing Boundary Conditions for the Wave Equation. Math. Comp., 1987, 49:65～90
[18] Higdon R L, Absorbing Boundary Conditions for Elastic Waves. Geophysics, 1991, 56(2):231～241
[19] Higdon R L, Absorbing Boundary Conditions for Acoustic and Elastic Waves in Stratified Media. J. Comp. Phys., 1992, 101:386～418
[20] Chew W. C. and Liu Q. H., Perfectly Matched Layers for Elastodynamics: A New Absorbing Boundary Condition. J. Comp. Acoust., Dec.1996, 4(4):72～79
[21] Yuan Xiaojuen, Borup D., Wiskin J.W., Berggren M., Eidens R., Johnson S.A., Formulation and validation of Berenger’s PML absorbing boundary for the FDTD simulation of acoustic scattering. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, July 1997, 44(4):816～822
[22] Quan Qi and Thomas L. Geers, Evaluation of the Perfectly Matched Layer for Computational Acoustics. J. Comp. Phys., 1998, 139:166～183
[23] Francis Collino and Chrysoula Tsogka, Application of the perfectly matched absorbing layer model to the linear elastodynamic problemin anisotropic heterogeneous media. Geophysics, January-February 2001, 66(1):294～307
[24] Mur G, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations, IEEE Tras. Electromagnetic Compatibility, 1981, 23:377～382
[25] Umashankar K. R. and A. Taflove, A novel method to analyze electromagnetic scattering of complex objects. IEEE Trans. Electromagnetic Compatibility, 1982, 24:397～405
[26] Schneider J. B., Wagner C. L., Ramahi O. M., Implementation of transparent sources in FDTD simulations. IEEE Trans. Antennas Propagat., Aug.1998, 46(8):1159～1168
[27] Schneider J. B., Wagner C. L. and Broschat S. L., Implementation of Transparent Sources Embedded in Acoustic Finite-Difference Time-Domain Grids. Journal of the Acoustical Society of America, 1998, 103(1):136～142
[28]王长清,祝西里编著,电磁场计算中的时域有限差分法,北京:北京大学出版社,1993.5
[29] C T, A Finite-Difference Model for Elastic Waves in the Ground. Diploma thesis, Univ. Braunschweig, Braunschweig, Germany, 1999 Schr& o&der
[30] Virieux J, SH-wave propagation in heterogenous media: Velocity-stress finite-difference method. Geophys., Nov.1984, 49: 1933～1957
[31] Virieux J., P-SV wave propagation in heterogenous media: Velociy-stress finite-difference method. Geophys., Apr.1986, 51:889～901
[32] Botteldooren D, Finite-difference time-domain simulation of low-frequency room acoustic problems. J. Acoust. Soc. Am., 1995, 98(6):3302～3308.
[33] Botteldooren D, Acoustical finite-difference time-domain simulation in a quasiCartesian grid. J. Acoust. Soc. Am., 1994, 95(5): 2313～2319.
[34] Maloney J. G. and Cummings K. E., Adaptation of FDTD techniques to acoustic modeling. in 11th Annual Review of Progress in Applied Computational Electronagnetics, Mar.1995, Vol.2, Monterey, CA: 724～731.
[35] 喻志远,时域有限差分计算中源的研究,电子科技大学学报,1998,27(1):43～46
[36] Yu Z., A Bandpass Source technique for the FDTD Analysis of Waveguide Discontinuity. Microw. Opt. Technol. Lett., 1998, 17(2):132～135
[37] 赵安平,于荣金,刘立杰,一种用于时域有限差分分析波导和微带电路的激励源技术,电子学报,1998,26(2):57～64
[38] Alterman Z. S. and F. C. Jr. Karal, Propagation of Elastic Waves in Layered Media by Finite-Difference Methods. Bull. Seism. Soc. Am., 1968, 58:367～398
[39] Vidale J E and Helmberger D V, Elastic Finite-Difference Modeling of the 1971 San Fernando, California Earthquake. Bull. Seism. Soc. Am., 1988, 78:122～141
[40] Helmberger D V and Vidale J E, Modeling Strong Motions Produced by Earthquakes with Two-Dimesional Numerical Codes. Bull. Seism. Soc. Am., 1988, 78:109～121
[41] Boore D M, Finite-Difference Methods for Seismic Wave Propagation in Heterogeneous Materials. In Vol. 11, Methods in Computational Physics (ed. Bolt, B. A.), Academic Press, New York, 1972
[42] Schr?der C T, Scott W R, On the Stability of the FDTD Algorithm for Elastic Media at a Material Interface [J]. IEEE Transactions on Geoscience and Remote Sensing, 2002, 40(2):474～481
[43] Jurgens T. G., A. Taflove, K. Umashankar, and T. G. Moore, Finite-difference time-domain modeling of curved surfaces, IEEE Trans. Antennas Propagat., Apr.1992, 40:357～366
[44] 廖振鹏著,工程波动理论导论,北京:科学技术出版社,2002.3
[45] Trefethen L N and Halpern L, Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions. Math. Comput., 1986, 47:421～435
[46] Halpern L and Trefethen L N, Wide-angle One-Way Wave Equations. J. Acoust. Soc. Am., 1988, 84:1397～1404
[47] Kolakowski H, Remarks on Perfectly Absorbing Boundary Conditions. Math. Meth. in the Appl. Sci., 1985, 7:486～492
[48] Kolakowski H, Absorbing Boundary Conditions for the Linearized Shallow Water Equations. Math. Meth. in the Appl. Sci., 1986, 8:41～49
[49] Halpern L and Rauch J, Error Analysis for Absorbing Boundary Conditions. Numer. Math., 1987, 51:459～467
[50] Higdon R L, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math. Comp., 1986, 47:437～459
[51] Keys R G, Absorbing Boundary Conditions for Acoustic Media. Geophys, 1985, 50:892～902
[52] Higdon R L, Numerical Absorbing Boundary Conditions for the Wave Equation. Math. Comp., 1987, 49:65～90
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