Application of a Perfectly Matched Layer to the Nonlinear Wave Equation

We derive a perfectly matched layer-like damping layer for the nonlinear wave equation. In the layer only two auxiliary variables are needed. In the linear case the layer is perfectly matched, but in the nonlinear case it is not. Well-posedness is established for the linear case. We also prove various energy estimates which can be used as a starting point for establishing stability of more general cases. In particular, we are able to show estimates for a special type of nonlinearity. Numerical experiments that show the effectiveness of the layer are presented both for nonlinear and linear problems. In the computations we use an eighth order summation-by-parts discretization in space and implement the boundary conditions using a penalty procedure. We present new stability results for this discretization applied to the second order wave equation in the case with Dirichlet boundary conditions.