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

• Published in: Journal of applied and industrial mathematics Vol. 7; no. 2; pp. 187 - 198
• Main Authors: Kabanikhin, S. I., Krivorot’ko, O. I.
• Format: Journal Article
• Language: English
• Published: Dordrecht SP MAIK Nauka/Interperiodica 01.04.2013; Springer Nature B.V
• Subjects: Algebra; Algorithms; Applied mathematics; Boundary conditions; Boundary value problems; Computational mathematics; Construction; Dirichlet problem; Inverse problems; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Numerical analysis; Regularization; Studies; Theorems; Wave equations; Dirichlet problem; degree of ill-posedness; singular value decomposition; wave equation
• ISSN: 1990-4789; 1990-4797
• DOI: 10.1134/S1990478913020075

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.