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

Contributed Talk

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Geometric Time Integration and absorbing boundary conditions. A case study

I. Alonso-Mallo and A. Portillo

Abstract
We are concerned about the confluence of two subjects of the numerical solution of time evolution PDEs: numerical methods that preserve geometric properties of the flow and absorbing boundary conditions used to reduce the computation to a finite domain. We pay attention to the time stability. The stability regions of time integrators are revisited. Since geometric methods are not always $A$-stable, it is necessary a suitable behavior of the real part of the eigenvalues of the spatially discretized problem to avoid in practice any time instability. The study is carried out for the one dimensional wave equation discretized with finite differences. Some numerical experiments confirm our results.

Bibliography
[1] I. Alonso-Mallo and A.M. Portillo, Absorbing boundary conditions and geometric integration: a case study for the wave equation, submitted.
[2] E. Hairer, E. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Second edition, Springer, Berlin, 2006.
[3] L. Halpern, Absorbing Boundary Conditions for the Discretization Schemes of the One-Dimensional Wave Equation, Math. Comput. 38 (1982), pp. 415-429.

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