Abstract: We construct an approximate Riemann solver for scalar advection–difusion equations with piecewise polynomial initial data. The objective is to handle advection and difusion simul- taneously to reduce the inherent numerical difusion produced by the usual advection fux 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 sub- regions. 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 defne the numerical fux within a fnite 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 fux calculation of advection and difusion. Also, it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efcient fnite volume scheme.

Key words: Generalized Riemann problem, Advection–difusion, Discontinuous Galerkin, Numerical fux, ADER, Difusive generalized Riemann problem, Space–time solution, Recovery method