An Adaptive Rothe Method for the Wave Equation

A b s r a c t he adaptive Rothe m e t h d approaches a time-dependent PDE as an ODE in funct in space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem thus perturbing the virtual function space method. The admissible size of that perturbatio can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individ-ually adapted spatial mesh. In this way, the method avods the wellkn difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagati of infor-mation. The present study concerns the adaptive Rothe method for hyper bolic equa t ins in the m d e l situation of the wave equation. All steps of the nstruction are given in detail and a numerical example (diffraction at a c rne r) is provided or the 2D wave equation. This example clearly indicates that the adaptive Rothe method i appropriate for problems which can gen-erally benefit from mesh adaptation. This should be even m r e p r u n c e d in the 3D case because of the strong Huygens ' principle