Absorbing boundary conditions and geometric integration: A case study for the wave equation

This paper is 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 the use of absorbing boundary conditions to reduce the computation to a finite domain. This confluence is studied with special attention to the time stability of the resulting full discretization. For this, the stability regions of the time integrators are revisited. Since geometric methods are not always AA-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. A deep study is carried out for the case of the one dimensional wave equation when it is discretized with finite differences, showing that this suitable behavior happens. Numerical experiments confirming the previous results are included.