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Abstract : |
Abstract. 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 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 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. Key words. Systems of conservation laws of mixed type, viscosity-capillarity admissibility criterion, the Lax-Friedrichs scheme, shock capturing schemes., |
