SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Geometric integration of two coupled wave equations with absorbing boundary conditions

A. Portillo and I. Alonso-Mallo

Abstract
Initial value problems with two coupled wave equations are considered. The problem is discretized in space using fourth order implicit finite differences. In order to reduce the computation to a bounded domain, absorbing boundary conditions are deduced. Well posed systems with fifth order of absorption for the diagonalizable case, and third order of absorption for the non diagonalizable one, are obtained. When a part of the solution reaches the boundary and it is absorbed, another part of the solution is still inside the computational interval, where it is important to preserve its geometric properties. This fact supports the simultaneous use of absorbing boundary conditions and symplectic integrators. Numerical experiments are displayed.

Bibliography
[1] I. Alonso-Mallo and A.M. Portillo, Geometric integration with absorbing boundary conditions: a case study for the wave equation, submitted.
[2] L. Halpern, Absorbing Boundary Conditions for the Discretization Schemes of the One-Dimensional Wave Equation, Math. Comput., 38 (1982), 415-429.
[3] L. J\`odar and D. Goberna, A matrix D'Alembert formula for coupled wave initial value problems, Computers Math Applic., 35 (1998), 1-15.
[4] S.K. Lele, Compact Finite Difference Schemes with Spectral-like Resolution J. Comput. Phys., 103 (1992), 16-42.

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