Error analysis of second-order locally implicit and local time-stepping 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 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 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.