Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.