Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows

• We present adaptive space–time discontinuous Galerkin method.
• The algorithm control the spatial, temporal and algebraic error.
• We present adaptive solution of non-stationary compressible Navier–Stokes equations.

We develop an adaptive numerical method for solution of the non-stationary compressible Navier–Stokes equations. This method is based on the space–time discontinuous Galerkin discretization, which employs high polynomial approximation degrees with respect to the space as well as to the time coordinates. We focus on the identification of the computational errors, following from the space and time discretizations and from the inexact solution of the arising nonlinear algebraic systems. We derive the residual-based error estimates approximating these errors. Then we propose an efficient algorithm which brings the algebraic, spatial and temporal errors under control. The computational performance of the proposed method is demonstrated by numerical experiments.