A High Order Space-time Riemann-solver-free Method for Solving Compressible Euler Equations

Abstract

In this paper, a novel high-order space-time method is introduced for solving compressible Euler equations. The method is inspired by two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The solution within each physical time step is updated alternately at the cell level and the vertex level. The arbitrarily high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each CE. The current method exhibits many advantageous features such as Riemannsolver-free, arbitrarily high-order accuracy, point-implicitness, and compactness. Several numerical tests will demonstrate the performance of the new scheme.