Dirichlet–Neumann alternating algorithm based on the natural boundary reduction for time-dependent problems over an unbounded domain

Abstract

In this paper, a new iterative algorithm to solve a time-dependent problem over an unbounded domain is suggested. This method is based on the natural boundary reduction and is suitable for solving initial boundary value problems of time-dependent wave equation over an unbounded domain. Firstly, an circular artificial boundary is introduced. Then the original unbounded domain is decomposed into a bounded domain and an exterior unbounded domain outside the artificial boundary. The natural integral equation obtained by the natural boundary reduction is used as a boundary condition on the artificial boundary. Secondly, a Dirichlet–Neumann (D–N) alternating iterative algorithm is constructed. The algorithm is equivalent to preconditioned Richardson iteration method. Thirdly, numerical studies are performed by finite element methods, and the results demonstrate the effectiveness of this algorithm. Finally, some remarks are presented.