Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem
HTML articles powered by AMS MathViewer
- by Olivier Vacus;
- Math. Comp. 74 (2005), 177-200
- DOI: https://doi.org/10.1090/S0025-5718-04-01669-2
- Published electronically: July 22, 2004
- PDF | Request permission
Abstract:
Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order—higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.References
- Alain Bamberger, Patrick Joly, and Jean E. Roberts, Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 323–352. MR 1043609, DOI 10.1137/0727021
- Francis Collino, High order absorbing boundary conditions for wave propagation models: straight line boundary and corner cases, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993) SIAM, Philadelphia, PA, 1993, pp. 161–171. MR 1227834
- F. Collino, Conditions absorbantes d’ordre élevé pour des modèles de propagation d’onde, technical report 1970, INRIA, 1993.
- 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
- Dan Givoli, Nonreflecting boundary conditions, J. Comput. Phys. 94 (1991), no. 1, 1–29. MR 1103713, DOI 10.1016/0021-9991(91)90135-8
- D. Givoli, Exact representations on artificial interfaces and applications in mechanics, Applied Mechanics Review, 52 (1999), pp. 333-349.
- 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 341888, DOI 10.1090/S0025-5718-1972-0341888-3
- T. Ha-Duong and P. Joly, On the stability analysis of boundary conditions for the wave equation by energy methods. I. The homogeneous case, Math. Comp. 62 (1994), no. 206, 539–563. MR 1216259, DOI 10.1090/S0025-5718-1994-1216259-2
- Thomas Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp. 47–106. MR 1819643, DOI 10.1017/S0962492900002890
- Bradley Alpert, Leslie Greengard, and Thomas Hagstrom, Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput. Phys. 180 (2002), no. 1, 270–296. MR 1913093, DOI 10.1006/jcph.2002.7093
- 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
- Semyon V. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math. 27 (1998), no. 4, 465–532. Absorbing boundary conditions. MR 1644674, DOI 10.1016/S0168-9274(98)00025-7
- P. Joly, O. Vacus, Stabilité de conditions aux limites pour l’équation des ondes par des méthodes énergétiques: le cas des bords courbes, INRIA Rocquencourt technical report 2849, France, 1996.
- Patrick Joly, Stéphanie Lohrengel, and Olivier Vacus, Un résultat d’existence et d’unicité pour l’équation de Helmholtz avec conditions aux limites absorbantes d’ordre 2, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 3, 193–198 (French, with English and French summaries). MR 1711059, DOI 10.1016/S0764-4442(00)88592-4
- Heinz-Otto Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 241010, DOI 10.1090/S0025-5718-1968-0241010-7
- 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, DOI 10.1090/S0025-5718-1986-0856695-2
- Dongwoo Sheen, Second-order absorbing boundary conditions for the wave equation in a rectangular domain, Math. Comp. 61 (1993), no. 204, 595–606. MR 1192975, DOI 10.1090/S0025-5718-1993-1192975-5
- O. Vacus, Singularités de frontières du domaine de calcul: le problème du coin, INRIA Rocquencourt technical report 2851, France, 1996.
Bibliographic Information
- Olivier Vacus
- Affiliation: CEA/CESTA, 33 114 Le Barp Cedex, France
- Email: vacus.olivier@wanadoo.fr
- Received by editor(s): May 26, 2002
- Received by editor(s) in revised form: June 29, 2003
- Published electronically: July 22, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 177-200
- MSC (2000): Primary 78A40, 65N12; Secondary 65M12, 46N40
- DOI: https://doi.org/10.1090/S0025-5718-04-01669-2
- MathSciNet review: 2085407