Volume 5, Issue 2
Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Wensheng Zhang, Li Tong & Eric T. Chung

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 205-228.

Published online: 2012-05

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  • Abstract

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis  shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms  based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.

  • Keywords

Acoustic wave equation implicit schemes ADI LOD stability condition dispersion curve MPI parallel computations

  • AMS Subject Headings

35L05 65M06 765Y05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-205, author = {Wensheng Zhang, Li Tong and Eric T. Chung}, title = {Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {2}, pages = {205--228}, abstract = {

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis  shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms  based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m1026}, url = {http://global-sci.org/intro/article_detail/nmtma/5935.html} }
TY - JOUR T1 - Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes AU - Wensheng Zhang, Li Tong & Eric T. Chung JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 205 EP - 228 PY - 2012 DA - 2012/05 SN - 5 DO - http://dor.org/10.4208/nmtma.2011.m1026 UR - https://global-sci.org/intro/article_detail/nmtma/5935.html KW - Acoustic wave equation KW - implicit schemes KW - ADI KW - LOD KW - stability condition KW - dispersion curve KW - MPI parallel computations AB -

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis  shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms  based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.

Wensheng Zhang, Li Tong & Eric T. Chung. (1970). Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes. Numerical Mathematics: Theory, Methods and Applications. 5 (2). 205-228. doi:10.4208/nmtma.2011.m1026
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