Volume 32, Issue 1
Stability for Imposing Absorbing Boundary Conditions in the Finite Element Simulation of Acoustic Wave Propagation

Wensheng Zhang, Eric T. Chung & Chaowei Wang

J. Comp. Math., 32 (2014), pp. 1-20.

Published online: 2014-02

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  • Abstract

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

  • Keywords

Stability Acoustic wave equation Simulation Finite element method Absorbing boundary conditions Wave operator decomposition

  • AMS Subject Headings

35L05 35L20 65M06 65M12 65M60.

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COPYRIGHT: © Global Science Press

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@Article{JCM-32-1, author = {}, title = {Stability for Imposing Absorbing Boundary Conditions in the Finite Element Simulation of Acoustic Wave Propagation}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {1}, pages = {1--20}, abstract = {

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1310-m3942}, url = {http://global-sci.org/intro/article_detail/jcm/9866.html} }
TY - JOUR T1 - Stability for Imposing Absorbing Boundary Conditions in the Finite Element Simulation of Acoustic Wave Propagation JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 20 PY - 2014 DA - 2014/02 SN - 32 DO - http://doi.org/10.4208/jcm.1310-m3942 UR - https://global-sci.org/intro/article_detail/jcm/9866.html KW - Stability KW - Acoustic wave equation KW - Simulation KW - Finite element method KW - Absorbing boundary conditions KW - Wave operator decomposition AB -

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

Wensheng Zhang, Eric T. Chung & Chaowei Wang. (1970). Stability for Imposing Absorbing Boundary Conditions in the Finite Element Simulation of Acoustic Wave Propagation. Journal of Computational Mathematics. 32 (1). 1-20. doi:10.4208/jcm.1310-m3942
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