Numerical Integrations of Systems of Conservation Laws of Mixed Type

The systems of conservation laws have been used to model dynamical phase transitions in, for example, the propagating phase boundaries in solids and the van der Waals uid. 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 a spike at the phase boundary which does not go away even with a reened 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 exibility in the ((rst order) scheme is used to obtain solutions for a wide range of the viscosity-capillarity equations.