Approximate Riemann Solvers and Numerical Flux Functions

Abstract

Given a monotone function $z(x)$ which connects two constant states, $u_L u_R )$, we find the unique (up to a constant) convex (concave) flux function, $\hat f(u)$, such that $z({x / t})$ is the physically correct solution to the associated Riemann problem. For $z({x / t})$, an approximate Riemann solver to a given conservation law, we derive simple necessary and sufficient conditions for it to be consistent with any entropy inequality. Associated with any member of a general class of consistent numerical fluxes, $h_f (u_R ,u_L )$, we have an approximate Riemann solver defined through $z(\zeta ) = ({{ - d} / {d_\zeta }})h_{f_\zeta } (u_R ,u_L )$, where $f_\zeta (u) = f(u) - \zeta u$. We obtain the corresponding $\hat f(u)$ via a Legendre transform and show that it is consistent with all entropy inequalities iff $h_{f_\zeta } (u_R ,u_L )$ is an E flux for each relevant $\zeta $. Examples involving commonly used two point numerical fluxes are given, as are comparisons with related work.