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On the accurate long-time solution of the wave equation in exterior domains: asymptotic expansions and corrected boundary conditions

Authors: Thomas Hagstrom, S. I. Hariharan and R. C. MacCamy
Journal: Math. Comp. 63 (1994), 507-539, S7
MSC: Primary 65N12; Secondary 35B40
MathSciNet review: 1248970
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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.

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  • [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1970.
  • [2] A. Barry, J. Bielak, and R. MacCamy, On absorbing boundary conditions for wave propagation, J. Comput. Phys. 79 (1988), 449-468. MR 973337 (90e:73046)
  • [3] A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (1980), 707-725. MR 596431 (82b:65091)
  • [4] J. Bielak and R. MacCamy, Dissipative boundary conditions for one-dimensional wave propagation, J. Integral Equations Appl. 2 (1990), 307-331. MR 1094472 (92d:35169)
  • [5] Y.-M. Chen, The transient behavior of diffraction of a plane pulse by a circular cylinder, Internat. J. Engrg. Sci. 2 (1964), 417-429. MR 0174251 (30:4458)
  • [6] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), 629-651. MR 0436612 (55:9555)
  • [7] B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math. 4 (1988), 21-45. MR 932317 (89c:65109)
  • [8] -, Long time behavior of absorbing boundary conditions, Math. Methods Appl. Sci. 13 (1990), 189-204. MR 1071439 (91m:35045)
  • [9] F.G. Friedlander, On the radiation field of pulse solutions of the wave equation, Proc. Roy. Soc. Ser. A 279 (1964), 386-394. MR 0164132 (29:1431)
  • [10] D. Givoli, Non-reflecting boundary conditions: A review, J. Comput. Phys. 94 (1991), 1-29.
  • [11] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Cambridge Univ. Press, Cambridge, 1990. MR 1050319 (91c:45003)
  • [12] B. Gustafsson, Inhomogeneous conditions at open boundaries for wave propagation problems, Appl. Numer. Math. 4 (1988), 3-19. MR 932316 (89b:65212)
  • [13] 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).
  • [14] T. Hagstrom, Consistency and convergence for numerical radiation conditions, Mathematical and Numerical Aspects of Wave Propagation Phenomena (G. Cohen, L. Halpern, and P. Joly, eds.), SIAM, Philadelphia, PA, 1991, pp. 283-292. MR 1106002
  • [15] T. Hagstrom and S.I. Hariharan, Far field expansion for anisotropic wave equations, Computational Acoustics: Scattering, Gaussian Beams and Aeroacoustics, vol. 2 (D. Lee, A. Cakmak, and R. Vichnevetsky, eds.), North-Holland, Amsterdam, 1990, pp. 283-294. MR 1095065 (92c:76056)
  • [16] T. Hagstrom and H.B. Keller, Exact boundary conditions at an artificial boundary for partial differential equations in cylinders, SIAM J. Math. Anal. 17 (1986), 322-341. MR 826697 (87g:35022)
  • [17] L. Halpern and J. Rauch, Error analysis for absorbing boundary conditions, Numer. Math. 51 (1987), 459-467. MR 902101 (88j:65195)
  • [18] S.I. Hariharan and R. MacCamy, Low frequency acoustic and electromagnetic scattering, Appl. Numer. Math. 2 (1986), 29-35. MR 834032 (87j:35095)
  • [19] R. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp. 47 (1986), 437-460. MR 856696 (87m:65131)
  • [20] S. Karp, A convergent 'far-field' expansion for two-dimensional radiation functions, Comm. Pure Appl. Math. 14 (1961), 427-434. MR 0135451 (24:B1500)
  • [21] J. Keller and D. Givoli, Exact non-reflecting boundary conditions, J. Comput. Phys. 82 (1989), 172-192. MR 1005207 (91a:76064)
  • [22] 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 0155091 (27:5033)
  • [23] 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 0223136 (36:6185)
  • [24] V.P. Mihailov, On the principle of limiting amplitude, Soviet Math. Dokl. 5 (1964), 1599-1602. MR 0170111 (30:352)
  • [25] L. Muravei, On the asymptotic behavior, for large values of the time, of solutions of exterior boundary value problems for the wave equation with two space variables, Math. USSR-Sb. 35 (1979), 377-423. MR 543811 (81i:35020)
  • [26] -, The wave equation and the Helmholtz equation in an unbounded domain with a star-shaped boundary, Proc. Steklov Inst. Math. 185 (1990), 191-201.
  • [27] R. Sakamoto, Hyperbolic boundary value problems, Cambridge Univ. Press, Cambridge, 1982. MR 666700 (83h:35071)
  • [28] V. Vladimirov, Yu. Drozhzhinov, and B. Zav'yalov, Tauberian theorems for generalized functions, Kluwer, Dordrecht, 1988. MR 947960 (89j:46043)
  • [29] D. Widder, The Laplace transform, Princeton Univ. Press, Princeton, NJ, 1946.

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Keywords: Boundary conditions, asymptotic expansions, wave equation
Article copyright: © Copyright 1994 American Mathematical Society

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