The role of the Jacobian in the adaptive discontinuous Galerkin method for the compressible Euler equations

Abstract

We provide a full description of the Jacobian to the discontinuous Galerkin discretisation of the compressible Euler equations, one of the key ingredients of the adaptive discontinuous Galerkin methods developed in previous publications. We demonstrate the use of this Jacobian within an implicit solver for the approximation of the (primal) stationary flow problems as well as in the adjoint (dual) problems that occur in the context of a posteriori error estimation and adaptive mesh refinement. In particular, we show that the (stationary) compressible Euler equations can efficiently be solved by the Newton method. Full quadratic Newton convergence is achieved on higher order elements as well as on locally refined meshes.