An Approximate Riemann Solver for Advection–Diffusion Based on the Generalized Riemann Problem

Abstract

We construct an approximate Riemann solver for scalar advection–diffusion equations with piecewise polynomial initial data. The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations. The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space–time discontinuous Galerkin approach with two subregions. The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem. In conjunction with a recovery polynomial, the Riemann solver is then applied to define the numerical flux within a finite volume method. Numerical results for a piecewise linear and a piecewise parabolic approximation are shown. These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion. Also, it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.