The HLL and HLLC Riemann Solvers

Abstract

For the purpose of computing a Godunov flux, Harten, Lax and van Leer [148] presented a novel approach for solving the Riemann problem approximately. The resulting Riemann solvers have become known as HLL Riemann solvers. In this approach an approximation for the intercell numerical flux is obtained directly, unlike the Riemann solvers presented previously in Chaps. 4 and 9. The central idea is to assume a wave configuration for the solution that consists of two waves separating three constant states. Assuming that the wave speeds are given by some algorithm, application of the integral form of the conservation laws gives a closed-form, approximate expression for the flux. The approach produced practical schemes after the contributions of Davis [94] and Einfeldt [105], who independently proposed various ways of computing the wave speeds required to completely determine the intercell flux. The resulting HLL Riemann solvers form the bases of very efficient and robust approximate Godunov-type methods. One difficulty with these schemes, however, is the assumption of a two-wave configuration. This is correct only for hyperbolic systems of two equations, such as the one-dimensional shallow water equations. For larger systems, such as the Euler equations or the split two-dimensional shallow water equations for example, the two-wave assumption is incorrect. As a consequence the resolution of physical features such as contact surfaces, shear waves and material interfaces, can be very inaccurate. For the limiting case in which these features are stationary relative to the mesh, the resulting numerical smearing is unacceptable. In view of these shortcomings of the HLL approach, a modification called the HLLC Riemann solver (C stands for Contact) was put forward by Toro, Spruce and Speares [347]. In spite of the limited experience available in using the HLLC scheme, the evidence is that this appears to offer a useful approximate Riemann solver for practical applications. Batten, Goldberg and Leschziner [18] have recently proposed implicit versions of the HLLC Riemann solver, and have applied the schemes to turbulent flows.