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High-order local absorbing conditions for the wave equation: Extensions and improvements. (English) Zbl 1136.65081

Summary: The solution of the time-dependent wave equation in an unbounded domain is considered. An artificial boundary \(\mathcal B\) is introduced which encloses a finite computational domain. On \(\mathcal B\) an absorbing boundary condition (ABC) is imposed. A formulation of local high-order ABCs recently proposed by T. Hagstrom and T. Warburton [Wave Motion 39, No. 4, 327–338 (2004; Zbl 1163.74364)] and based on a modification of the ABCs of R. L. Higdon [Math. Comput. 49, 65–90 (1987; Zbl 0654.65083); SIAM J. Numer. Anal. 31, No. 1, 64–100 (1994; Zbl 0798.65113)], is further developed and extended in a number of ways.
First, the ABC is analyzed in new ways and important information is extracted from this analysis. Second, the ABCs are extended to the case of a dispersive medium, for which the Klein-Gordon wave equation governs. Third, the case of a stratified medium is considered and the way to apply the ABCs to this case is explained. Fourth, the ABCs are extended to take into account evanescent modes in the exact solution. The analysis is applied throughout this paper to two-dimensional wave guides.
Two numerical algorithms incorporating these ABCs are considered: a standard semi-discrete finite element formulation in space followed by time-stepping, and a high-order finite difference discretization in space and time. Numerical examples are provided to demonstrate the performance of the extended ABCs using these two methods.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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