Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems

Abstract

Phase transition problems in compressible media can be modelled by mixed hyperbolic-elliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. For numerical approximation tracking-type algorithms have been suggested. The core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel subiteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. To our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases.