Abstract:We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study both the short- and long-time behavior of the error. It is proved that, in two space dimensions, no local in time, constant-coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable-coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant-coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions, using energy methods and leading to asymptotically correct error bounds.
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1970.
- A. Barry, J. Bielak, and R. C. MacCamy, On absorbing boundary conditions for wave propagation, J. Comput. Phys. 79 (1988), no. 2, 449–468. MR 973337, DOI 10.1016/0021-9991(88)90025-3
- Alvin Bayliss and Eli Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (1980), no. 6, 707–725. MR 596431, DOI 10.1002/cpa.3160330603
- J. Bielak and R. C. MacCamy, Dissipative boundary conditions for one-dimensional wave propagation, J. Integral Equations Appl. 2 (1990), no. 3, 307–331. MR 1094472, DOI 10.1216/jiea/1181075566
- Yung Ming Chen, The transient behavior of diffraction of plane pulse by a circular cylinder, Internat. J. Engrg. Sci. 2 (1964), 417–429 (English, with French, German, Italian and Russian summaries). MR 0174251, DOI 10.1016/0020-7225(64)90020-5
- Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI 10.1090/S0025-5718-1977-0436612-4
- Björn Engquist and Laurence Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math. 4 (1988), no. 1, 21–45. MR 932317, DOI 10.1016/S0168-9274(88)80004-7
- B. Engquist and L. Halpern, Long-time behaviour of absorbing boundary conditions, Math. Methods Appl. Sci. 13 (1990), no. 3, 189–203. MR 1071439, DOI 10.1002/mma.1670130302
- F. G. Friedlander, On the radiation field of pulse solutions of the wave equation. II, Proc. Roy. Soc. London Ser. A 279 (1964), 386–394. MR 164132, DOI 10.1098/rspa.1964.0111 D. Givoli, Non-reflecting boundary conditions: A review, J. Comput. Phys. 94 (1991), 1-29.
- G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
- Bertil Gustafsson, Inhomogeneous conditions at open boundaries for wave propagation problems, Appl. Numer. Math. 4 (1988), no. 1, 3–19. MR 932316, DOI 10.1016/S0168-9274(88)80003-5 T. Ha-Duong and P. Joly, On the stability analysis of boundary conditions for the wave equation by energy methods, Part I: the homogeneous case, INRIA Rapports de Recherche 1306, (1990).
- Thomas Hagstrom, Consistency and convergence for numerical radiation conditions, Mathematical and numerical aspects of wave propagation phenomena (Strasbourg, 1991) SIAM, Philadelphia, PA, 1991, pp. 283–292. MR 1106002
- S. I. Hariharan and Thomas Hagstrom, Far field expansion for anisotropic wave equations, Computational acoustics, Vol. 2 (Princeton, NJ, 1989) North-Holland, Amsterdam, 1990, pp. 283–294. MR 1095065
- Thomas Hagstrom and H. B. Keller, Exact boundary conditions at an artificial boundary for partial differential equations in cylinders, SIAM J. Math. Anal. 17 (1986), no. 2, 322–341. MR 826697, DOI 10.1137/0517026
- Laurence Halpern and Jeffrey Rauch, Error analysis for absorbing boundary conditions, Numer. Math. 51 (1987), no. 4, 459–467. MR 902101, DOI 10.1007/BF01397547
- S. I. Hariharan and R. C. MacCamy, Low frequency acoustic and electromagnetic scattering, Appl. Numer. Math. 2 (1986), no. 1, 29–35. MR 834032, DOI 10.1016/0168-9274(86)90012-7
- Robert L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp. 47 (1986), no. 176, 437–459. MR 856696, DOI 10.1090/S0025-5718-1986-0856696-4
- S. N. Karp, A convergent ’farfield’ expansion for two-dimensional radiation functions, Comm. Pure Appl. Math. 14 (1961), 427–434. MR 135451, DOI 10.1002/cpa.3160140318
- Joseph B. Keller and Dan Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys. 82 (1989), no. 1, 172–192. MR 1005207, DOI 10.1016/0021-9991(89)90041-7
- P. D. Lax, C. S. Morawetz, and R. S. Phillips, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math. 16 (1963), 477–486. MR 155091, DOI 10.1002/cpa.3160160407
- C. S. Morawetz and D. Ludwig, An inequality for the reduced wave operator and the justification of geometrical optics, Comm. Pure Appl. Math. 21 (1968), 187–203. MR 223136, DOI 10.1002/cpa.3160210206
- V. P. Mihaĭlov, On the principle of limiting amplitude, Dokl. Akad. Nauk SSSR 159 (1964), 750–752 (Russian). MR 0170111
- L. A. Muraveĭ, Asymptotic behavior for large values of time of the solutions of exterior boundary problems for the wave equation with two space variables, Proceedings of an All-Union Conference on Partial Differential Equations (Moscow State Univ., Moscow, 1976) Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 1978, pp. 165–169 (Russian). MR 543811 —, The wave equation and the Helmholtz equation in an unbounded domain with a star-shaped boundary, Proc. Steklov Inst. Math. 185 (1990), 191-201.
- Reiko Sakamoto, Hyperbolic boundary value problems, Cambridge University Press, Cambridge-New York, 1982. Translated from the Japanese by Katsumi Miyahara. MR 666700
- V. S. Vladimirov, Yu. N. Drozzinov, and B. I. Zavialov, Tauberian theorems for generalized functions, Mathematics and its Applications (Soviet Series), vol. 10, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR 947960, DOI 10.1007/978-94-009-2831-2 D. Widder, The Laplace transform, Princeton Univ. Press, Princeton, NJ, 1946.
- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp. 63 (1994), 507-539
- MSC: Primary 65N12; Secondary 35B40
- DOI: https://doi.org/10.1090/S0025-5718-1994-1248970-1
- MathSciNet review: 1248970