High-Resolution Finite Volume Central Schemes for a Compressible Two-Phase Model

Abstract

A modification of the Kurganov, Noelle, Petrova central-upwind scheme [A. Kurganov et al., SIAM J. Sci. Comput., 23 (2001), pp. 707--740] for hyperbolic systems of conservation laws is presented. In this work, the numerical scheme is applied to a single-temperature model for compressible two-phase flow with pressure and velocity relaxations [E. Romenski et al., J. Sci. Comput., 42 (2010), pp. 68--95]. The system of governing equations of this model is expressed in conservative form, which is the necessary condition to use a central scheme. The numerical scheme presented is based not on the complete characteristic decomposition but only on the information about the local speeds of propagation given by the maximum and minimum eigenvalues of the Jacobian of the fluxes. We propose to use the numerical flux formulation of the central-upwind scheme in conjunction with a second-order reconstruction of the primitive variables and the MUSCL-Hancock method, where the boundary-extrapolated values are evolved by half a time step before the computation of the numerical fluxes. To investigate the accuracy and robustness of the proposed scheme, two 1D Riemann problems of an air/water mixture and a 2D shock-bubble interaction problem are presented. Furthermore, a detailed comparison of the proposed scheme with the second-order GFORCE scheme and the first-order Lax--Friedrichs scheme is shown. To integrate the source terms, an operator-splitting approach is used, and, under suitable conditions, it is shown that this integration can be computed analytically.