## Dirichlet-Neumann alternating algorithm based on the natural boundary reduction for time-dependent problems over an unbounded domain.(English)Zbl 1013.65102

Summary: 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.

### MSC:

 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 35L05 Wave equation
Full Text:

### References:

  Givoli, D., Numerical methods for problems in infinite domains, (1992), Elsevier Amsterdam · Zbl 0788.76001  Feng, K., Asymptotic radiation conditions for reduced wave equations, J. comput. math., 2, 2, 130-138, (1984) · Zbl 0559.65085  M.J. Grote, Non-reflecting boundary conditions. Ph.D. Dissertation, Stanford University, Stanford, CA, 1995  Grote, M.J.; Keller, J.B., Exact non-reflecting boundary conditions for the dependent wave equation, SIAM J. appl. math., 55, 2, 280-297, (1995) · Zbl 0817.35049  Grote, M.J.; Keller, J.B., Non-reflecting boundary conditions for time dependent scattering problems, J. comput. phys., 127, 1, 52-65, (1996) · Zbl 0860.65080  Huan, R.; Thompson, L.L., Accurate radiation boundary conditions for the time-dependent wave equation on unbounded domain, Internat. J. numer. methods engrg., 47, 9, 1569-1603, (2000) · Zbl 0965.65120  Thompson, K.W., Time dependent boundary conditions for nonlinear hyperbolic systems, J. comput. phys., 61, 171-214, (1992)  Thompson, K.W., Time dependent boundary conditions for hyperbolic systems, II, J. comput. phys., 89, 439-461, (1990) · Zbl 0701.76070  Yu, Dehao, A domain decomposition method based on natural boundary reduction over unbounded domain, Math. numer. sinica, 16, 4, 448-459, (1994), Chinese J. Numer. Math. Appl. 7 (1) (1995) 95-105 · Zbl 0900.65340  Yu, Dehao, Discretization of non-overlapping domain decomposition method for unbounded domains and its convergence, Math. numer. sinica, 18, 3, 328-336, (1996), Chinese J. Numer. Math. Appl. 18 (4) (1996) 93-102 · Zbl 0880.65092  Feng, Kang; Yu, Dehao, Natural boundary reduction and domain decomposition, (), 367-371  G.C. Hsiao, B.N. Khoromskij, W.L. Wendland, Boundary integral operators and domain decomposition, Preprint 94-11, Mathematishes Institut A, Universität Stuttgart, 1994  Du, Qikui; Yu, Dehao, Natural integral equation of parabolic initial boundary value problem and its numerical implementation, Chinese J. numer. math. appl., 22, 1, 88-101, (2000) · Zbl 0960.65109  Du, Qikui; Yu, Dehao, On the natural integral equation for two dimensional hyperbolic equation, Acta math. appl. sinica, 24, 1, 17-26, (2001) · Zbl 0974.35071  Yu, Dehao, Mathematical theory of natural boundary element method, (1993), Science Press Beijing
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.