A nonlinear approach to absorbing boundary conditions for the semilinear wave equation

Szeftel, Jérémie

doi:10.1090/S0025-5718-06-01820-5

A nonlinear approach to absorbing boundary conditions for the semilinear wave equation
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Math. Comp. 75 (2006), 565-594 Request permission

Abstract:

We construct a family of absorbing boundary conditions for the semilinear wave equation. Our principal tool is the paradifferential calculus which enables us to deal with nonlinear terms. We show that the corresponding initial boundary value problems are well posed. We finally present numerical experiments illustrating the efficiency of the method.

References

Ali Alabidi, Réflexion transverse des singularités pour un problème aux limites non linéaire d’ordre $2$, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 10, 287–290 (French, with English summary). MR 786901
Bradley Alpert, Leslie Greengard, and Thomas Hagstrom, Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal. 37 (2000), no. 4, 1138–1164. MR 1756419, DOI 10.1137/S0036142998336916
Jean-Pierre Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185–200. MR 1294924, DOI 10.1006/jcph.1994.1159
Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751, DOI 10.24033/asens.1404
Jean-Yves Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995), 177 (French, with French summary). MR 1340046
E. Dubach, Nonlinear artificial boundary conditions for the viscous Burgers equation, prépublication 00/04 de l’université de Pau et des pays de l’Adour, 2000.
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, DOI 10.1002/cpa.3160320303
Marcus J. Grote and Joseph B. Keller, Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math. 55 (1995), no. 2, 280–297. Perturbation methods in physical mathematics (Troy, NY, 1993). MR 1322761, DOI 10.1137/S0036139993269266
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
L. Halpern and J. Rauch, Absorbing boundary conditions for diffusion equations, Numer. Math. 71 (1995), no. 2, 185–224 (English, with English and French summaries). MR 1347164, DOI 10.1007/s002110050141
Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York, Inc., New York, 1969. Third revised printing. MR 0248435, DOI 10.1007/978-3-662-30722-9
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
E. L. Lindmann, Free-space boundary conditions for the time dependent wave equation, J. Comp. Phys. 18 (1975), 16–78.
Andrew Majda and Stanley Osher, Reflection of singularities at the boundary, Comm. Pure Appl. Math. 28 (1975), no. 4, 479–499. MR 492792, DOI 10.1002/cpa.3160280404
Yves Meyer, Remarques sur un théorème de J.-M. Bony, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 1–20 (French). MR 639462
Louis Nirenberg, Lectures on linear partial differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972. MR 0450755, DOI 10.1090/cbms/017
Monique Sablé-Tougeron, Régularité microlocale pour des problèmes aux limites non linéaires, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 1, 39–82 (French). MR 840713, DOI 10.5802/aif.1037
Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
Jérémie Szeftel, Absorbing boundary conditions for reaction-diffusion equations, IMA J. Appl. Math. 68 (2003), no. 2, 167–184. MR 1968310, DOI 10.1093/imamat/68.2.167
Jérémie Szeftel, Design of absorbing boundary conditions for Schrödinger equations in $\Bbb R^d$, SIAM J. Numer. Anal. 42 (2004), no. 4, 1527–1551. MR 2114289, DOI 10.1137/S0036142902418345
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
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954