A Riemann Problem Based Method for the Resolution of Compressible Multimaterial Flows

A correction for Godunov-type methods is described, yielding a perfect capture of contact discontinuities, in hydrodynamic flow regime. The correction is based upon a simple idea: starting from a nondegraded solution at a given instant, the use of an Eulerian scheme around a contact discontinuity will entail, at the next instant, the degradation of the solution at only the two adjacent nodes to the discontinuity. The exact solution of the Riemann problem yields the state variables on both sides of the discontinuity. Knowledge of these variables may be used to correct the two nodes affected by numerical diffusion. The method is applied to problems involving a gas?liquid interface. The liquid is assumed compressible, obeying the “stiffened gas” equation of state, for which the solution of the Riemann problem is easily obtained. The method is first tested with 1D problems which have either an exact solution or accurate numerical solutions in the literature. Then the concept is extended in two dimensions. Assuming that the 1D Riemann problem along the normal to the interface is a reasonable approximation of the 2D Riemann problem for Euler equations, we extend efficiently the algorithm for two-dimensional interface problems. Several two-dimensional test cases are presented for which the method provides accurate solutions.