Intro

The Riemann problem is an IVP for the Euler Equations, which consists of 2 constant initial data states separated by a discontinuity between them, say at x=0. For x<x=0, we say the data state is XL, for which there are initial variables WL=(rho_L, p_L, u_L). Similarly for the initial right data state, XR=x>x=0, the initial variables are WR=(rho_R, p_R, u_R). The solution of this IVP consists of 3 nonlinear waves, a left wave that is either a shock or a rarefaction, a center contact discontinuity, and a right wave that is either a shock or a rarefaction wave. Depending on which type of L- or R-wave is present, different expressions exist that describe the change in variables across them. The contact discontinuity is special in that the pressure (p) and velocity (u) are constant across it.

Program Outline for Exact Riemann Solver

Given the above properties of the Riemann problem, there is an algebraic expression of which the root is the p in the central region or "star-region" (denoted by '*'), between the L- and R- waves. The overall structure of the Riemann problem then is to solve this algebraic equation for p*. Once p* is known, u* follows immediately. The remaining rho*_L and rho*_R follow from expressions valid for the specific L- or R- wave present.