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

Abstract

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem Aq = f with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator A in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem Aq = f on a uniform grid is reduced to a system of linear algebraic equations A ll q = F. Using the singular value decomposition, we find singular values of the matrix A ll and develop a numerical algorithm for constructing the r-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated r-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.