An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations

In this paper we propose a discontinuous Galerkin scheme for the numerical approximation of unsteady heat conduction and diffusion problems in multi dimensions. The scheme is based on a discrete space–time variational formulation and uses an explicit approximative solution as predictor. This predictor is obtained by a Taylor expansion about the barycenter of each grid cell at the old time level in which all time or mixed space–time derivatives are replaced by space derivatives using the differential equation several times. The heat flux between adjacent grid cells is approximated by a local analytical solution. It takes into account that the approximate solution may be discontinuous at grid cell interfaces and allows the approximation of discontinuities in the heat conduction coefficient. The presented explicit scheme has to satisfy a typical parabolic stability restriction. The loss of efficiency, especially in the case of strongly varying sizes of cells in unstructured grids, is circumvented by allowing different time steps in each grid cell which are adopted to the local stability restrictions. We discuss the linear stability properties in this case of varying diffusion coefficients, varying space increments and local time steps and extent these considerations also to a modified symmetric interior penalization scheme. In numerical simulations we show the efficiency and the optimal order of convergence in space and time.