Positivity-preserving flux-limited method for compressible fluid flow

Abstract

The extension of a flux discretization method to second-order accuracy can lead to some difficulties in maintaining positivity preservation. While the MUSCL-TVD scheme maintains the positivity preservation property of the underlying 1st-order flux discretization method, a flux-limited-TVD scheme does not. A modification is here proposed to the flux-limited-TVD scheme to make it positivity-preserving when used in conjunction with the Steger-Warming flux vector splitting method. The proposed algorithm is then compared to MUSCL for several test cases. Results obtained indicate that while the proposed scheme is more dissipative in the vicinity of contact discontinuities, it performs significantly better than MUSCL when solving strong shocks in hypersonic flowfields: the amount of pressure overshoot downstream of the shock is minimized and the time step can be set to a value typically two or three times higher. While only test cases solving the one-dimensional Euler equations are here presented, the proposed scheme is written in general form and can be extended to other physical models.