A Runge-Kutta based Discontinuous Galerkin Method with Time Accurate Local Time Stepping

An explicit one-step time discretization for discontinuous Galerkin schemes applied to advection-diffusion equations is presented. The main idea is based on a predictor corrector approach which was proposed in [1]. The interesting feature is that the predictor is local and takes only into account the time evolution of the data within each grid cell. In this paper, we focus ourselves to a continuous extension Runge-Kutta schemes which was described in [2]. The advantage of the predictor corrector formulation is that the time evolution is done in one step and the data of the direct neighbors are needed only. Hence, the proposed discontinuous Galerkin scheme has the optimal locality within the whole time step. This is the basis to introduce a timeconsistent local time stepping in a way such that every grid cell may run with its own optimal time step as given by the local stability restriction [3]. The time accuracy and the efficiency of the local time stepping is shown for linear and non-linear problems. A direct numerical simulation of the aeroacoustics of a natural gas injector shows the efficiency of the presented methodology, being well suited for unsteady advection dominated problems with adaptive schemes.