Error control for a class of Runge-Kutta discontinuous Galerkin methods for nonlinear conservation laws

We propose an a posteriori error estimate for the Runge-Kutta discontinuous Galerkin method (RK-DG) of arbitrary order in arbitrary space dimensions. For stabilization of the scheme a general framework of projections is introduced. Finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy. Numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids.