Numerical Integrations of Systems of Conservation Laws of Mixed Type

The systems of conservation laws have been used to model dynamic-phase transitions in, for example, the propagating-phase boundaries in solids and the van der Waals fluid. When integrating such mixed hyperbolic-elliptic systems the Lax–Friedrichs scheme is known to give the correct solutions selected by a viscosity-capillarity criterion, except for a spike at the phase boundary which does not go away even with a refined mesh [15]. We identify the source of this spike as an inconsistency between the Lax–Friedrichs discretization and the viscosity-capillarity equations, and show a simple change of variable that can eliminate this spike. We then implement a high-resolution scheme for the mixed-type problems that select the same viscosity-capillarity solutions as the Lax–Friedrichs scheme with higher resolutions. Furthermore, a flexibility in the (first-order) scheme is used to obtain solutions for a wide range of the viscosity-capillarity equations.