Building a van Leer-type numerical scheme for a model of two-phase flows

摘要

A van Leer-type numerical scheme for a model of two-phase flows is constructed. The governing equations were derived from the modeling of deflagration-to-detonation transitions in granular materials. The system contains source terms in nonconservative form, which cause lots of inconveniences for standard numerical schemes. Our proposed scheme is relied on exact solutions of local Riemann problems. Then, we provide many numerical tests, in which the errors and orders of accuracy of this scheme are computed. These tests show that our proposed van Leer-type scheme has a much better accuracy than the Godunov-type scheme, and that the scheme is well-balanced in the sense that it can capture exactly stationary waves. Furthermore, comparisons between van Leer’s limiter and Roe’s superbee limiter are given.