High resolution finite volume central schemes for a compressibile two-phase model

Abstract

A modi_cation 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 ow 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 are expressed in conservative form, which is the necessary condition to use a central scheme. The numerical scheme presented is not based on the complete characteristic decomposition, but only on the information about the local speeds of propagation given by the maximum and minimum eigenvalue of the Jacobian of the uxes. We propose to use the numerical ux 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 time step before the computation of the numerical uxes. 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 with the second order GFORCE scheme and the _rst 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.