A Novel Solver for the Generalized Riemann Problem Based on a Simplified LeFloch–Raviart Expansion and a Local Space–Time Discontinuous Galerkin Formulation

Abstract

In a wide class of high order shock-capturing methods for hyperbolic conservation laws, the solution of the conservation law is represented at each time-step by a piecewise smooth function (say, a polynomial reconstructed from cell-averages or an approximation in a finite element space). To maintain a sharp resolution of shock waves, jumps at the cell boundaries are allowed. The resulting initial value problem with piecewise smooth but discontinuous initial data is called the generalized Riemann problem. We present a new solver for the generalized Riemann problem based on a simplified version of a local asymptotic series expansion constructed by LeFloch and Raviart (Ann Inst H Poincare Anal Non Lineaire 5:179---207, 1988). Contrary to the original approach, in our new solver no higher order flux derivatives and other nonlinear terms need to be computed. Moreover, we introduce a new variant of the local space---time DG method of Dumbser et al. (J Comput Phys 227:3971---4001, 2008), that allows us to use a direct solution strategy for the generalized Riemann problem without relying on a Cauchy---Kovalevskaya procedure for the flux computation.