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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.