Chapter 2 - The Riemann Problem: Solvers and Numerical Fluxes

Though introduced by Bernhard Riemann more than 150 years ago (Riemann, 1860), the Riemann problem entered the field of modern computational science, with the pioneering work of Godunov (1959), almost a century later. Here, the Riemann problem is first defined and illustrative examples are given, before providing essential background on hyperbolic equations for later use. The exact solution of the Riemann problem for the compressible Euler equation, the canonical hyperbolic system, is then presented in some detail. This sets the bases for studying and critically assessing approximate solution methods, such as the Roe solver, HLL (Harten, Lax, van Leer), HLLC (Harten, Lax, van Leer, Contact) and an Osher–Solomon type solver. Uses of the Riemann problem solution are discussed, starting from its role in defining numerical fluxes for finite volume and discontinuous Galerkin finite element methods. Alternative approaches for defining numerical fluxes are also touched upon, including flux vector splitting, centred and multistage type fluxes, such as MUlti-STAge (MUSTA), Krylov type and PVM (polynomial viscosity matrix) schemes. Criteria to judge existing Riemann solvers and related concepts are discussed, along with relevant references for further study. Some possible generalisations of the classical Riemann problem are mentioned, notably multidimensional Riemann solvers and the high-order or generalised Riemann problem.