A Compact High Order Space-Time Method for Conservation Laws

Abstract

This paper presents a novel high-order space-time method for hyperbolic conservation laws. 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, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

Accuracy Enhancement of a Riemann-Solver-Free Spacetime Discontinuous Galerkin Method via Constrained Least Square Reconstruction

“…Without loss of generality, a source term r is also included in Eq. (1). Note that f may contain both advective and di↵usive fluxes.…”

“…[1][2][3] The method was inspired by the well-known Conservation Element/Solution Element (CE/SE) method 4 and the discontinuous Galerkin (DG) method. 5 The method adopts the concepts of using staggered spacetime meshes to enforce spacetime flux conservation (cf.…”

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

“…Due to the cell-vertex solution updating strategy and its DG ingredient, the current method has been termed as the DG-CVS method [4].…”

“…The idea is to use the divergence theorem explained in [17] to reduce the volume integral to surface integrals and further to line integrals. The detailed procedure has been described in [4] and will not be repeated here.…”

“…However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Due to the alternate cell-vertex solution updating strategy and its DG ingredient, the method has been termed as the discontinuous Galerkin cellvertex scheme (DG-CVS) [4].…”