Numerical absorbing boundary conditions for the wave equation

Author:
Robert L. Higdon

Journal:
Math. Comp. **49** (1987), 65-90

MSC:
Primary 65N05; Secondary 35L05

MathSciNet review:
890254

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Abstract | References | Similar Articles | Additional Information

Abstract: We develop a theory of difference approximations to absorbing boundary conditions for the scalar wave equation in several space dimensions. This generalizes the work of the author described in [8]. The theory is based on a representation of analytical absorbing boundary conditions proven in [8]. These conditions are defined by compositions of first-order, one-dimensional differential operators. Here the operators are discretized individually, and their composition is used as a discretization of the boundary condition. The analysis of stability and reflection properties reduces to separate studies of the individual factors. A representation of the discrete boundary conditions makes it possible to perform the analysis geometrically, with little explicit calculation.

**[1]**A. Bamberger, B. Engquist, L. Halpern & P. Joly,*Construction et Analyse d'Approximations Paraxiales en Milieu Hétérogène*. I*and*II, Internal Reports 114 and 128, Centre de Mathématiques Appliquées, Ecole Polytechnique, 1984-1985.**[2]**Alvin Bayliss and Eli Turkel,*Radiation boundary conditions for wave-like equations*, Comm. Pure Appl. Math.**33**(1980), no. 6, 707–725. MR**596431**, 10.1002/cpa.3160330603**[3]**Bjorn Engquist and Andrew Majda,*Absorbing boundary conditions for the numerical simulation of waves*, Math. Comp.**31**(1977), no. 139, 629–651. MR**0436612**, 10.1090/S0025-5718-1977-0436612-4**[4]**Björn Engquist and Andrew Majda,*Radiation boundary conditions for acoustic and elastic wave calculations*, Comm. Pure Appl. Math.**32**(1979), no. 3, 314–358. MR**517938**, 10.1002/cpa.3160320303**[5]**Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström,*Stability theory of difference approximations for mixed initial boundary value problems. II*, Math. Comp.**26**(1972), 649–686. MR**0341888**, 10.1090/S0025-5718-1972-0341888-3**[6]**Laurence Halpern and Lloyd N. Trefethen,*Wide-angle one-way wave equations*, J. Acoust. Soc. Amer.**84**(1988), no. 4, 1397–1404. MR**965847**, 10.1121/1.396586**[7]**Robert L. Higdon,*Initial-boundary value problems for linear hyperbolic systems*, SIAM Rev.**28**(1986), no. 2, 177–217. MR**839822**, 10.1137/1028050**[8]**Robert L. Higdon,*Absorbing boundary conditions for difference approximations to the multidimensional wave equation*, Math. Comp.**47**(1986), no. 176, 437–459. MR**856696**, 10.1090/S0025-5718-1986-0856696-4**[9]**E. L. Lindman, ""Free-space" boundary conditions for the time dependent wave equation,"*J. Comput. Phys.*, v. 18, 1975, pp. 66-78.**[10]**Lloyd N. Trefethen,*Instability of difference models for hyperbolic initial-boundary value problems*, Comm. Pure Appl. Math.**37**(1984), no. 3, 329–367. MR**739924**, 10.1002/cpa.3160370305**[11]**Lloyd N. Trefethen and Laurence Halpern,*Well-posedness of one-way wave equations and absorbing boundary conditions*, Math. Comp.**47**(1986), no. 176, 421–435. MR**856695**, 10.1090/S0025-5718-1986-0856695-2**[12]**Guan Quan Zhang,*High order approximation of one-way wave equations*, J. Comput. Math.**3**(1985), no. 1, 90–97. MR**815413**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890254-1

Keywords:
Absorbing boundary conditions,
wave equation,
initial-boundary value problems

Article copyright:
© Copyright 1987
American Mathematical Society