A numerical method for solving the Dirichlet problem for the wave equation

S. I. Kabanikhin ; O. I. Krivorot'ko

Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 4, pp. 90-101.

Résumé

A numerical method for solving the Dirichlet problem for the wave equation in the two-dimensional space is constructed. An analysis of the ill-posedness of the problem is carried out and a reguralization algorthm is constructed. The first step in the regularization of the problem consists in expansion in a Forier series with respect to one of the variables and passage to a finite sequence of Dirichlet problems for the wave equation in the one-dimensional space. Each of the Dirichlet problems obtained for the wave equation in the one-dimensional space is reduced to the inverse problem $Aq=f$ to some direct (correct) problem. We accomplish an analysis of the ill-posedness degree of the inverse problem on the basis of the study of the nature of the decay of the singular values of $A$ and its discrete analog $A_{mn}$. For relatively small values $m$ and $n$, we develop a numerical algorithm for constructing $r$-solutions to the inverse problem. For the general case, we apply an optimization method for solving the inverse problem. The results of numerical calculations are given.

Détail

Keywords: Dirichlet problem, wave equation, ill-posedness degree
Mots-clés : singular value decomposition.

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