On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

Abstract

A posteriori bounds for the error measured in various norms for a standard second-order explicit-in-time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one-dimensional (in space) linear transport problem are derived. The proof is based on a novel space-time polynomial reconstruction, hinging on high-order temporal reconstructions for continuous and discontinuous Galerkin time-stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.