Exact Riemann solvers for conservation laws with phase change

Abstract

In this paper we consider the solid–solid phase transformation in martensitic materials and present two numerical procedures for solving exactly the Riemann problems of a 3×3 system of conservation laws [21]. A particular attention is given to the configurations of the phase boundaries. For a Riemann problem whose initial states are specified in different phases, we first assume that the phase boundary is stationary and then find the solution through an iteration method [24]. Configuration of the transition front is then determined based on this stationary-phase-boundary solution [12]. The solution with dynamic phase change is calculated by listing all the relations in the Riemann problem and solving the resulting nonlinear system. Another approach, which avoids solving this system, is also proposed where the solution is obtained by computing the intersection of two projection curves. A front capturing/tracking method [25] implementing these Riemann solvers is presented to approximate initial value problems with propagating transition fronts. This approach captures the phase boundary sharply without artificial smearing in the physically unstable region.