High order full discretizations of coupled wave equations with absorbing boundary conditions and geometric integration

Coupled wave initial value problems are considered. First, problems are spatially discretized by means of fourth order implicit finite differences. To obtain numerical approximations in a bounded domain, local absorbing boundary conditions are deduced. Well posedness is studied for systems with fifth order of absorption for the diagonalizable case and third order of absorption for the non-diagonalizable one. The solution of these coupled problem can stay long time inside the computational window so that the use of geometric time integrators is justified. In particular, the behavior of a symplectic time integrator is studied, showing advantages and disadvantages of its use. Numerical experiments that confirm our results are displayed.