Journal of the
Korean Mathematical Society JKMS

We present a multi-stage Roe-type numerical scheme for a model of two-phase flows arisen from the modeling of deflagration-to-detonation transition in granular materials. The first stage in the construction of the scheme computes the volume fraction at every time step. The second stage deals with the nonconservative terms in the governing equations which produces states on both side of the contact wave at each node.In the third stage, a Roe matrix for the two-phase is used to apply on the states obtained from the second stage. This scheme is shown to capture stationary waves and preserves the positivity of the volume fractions. Finally, we present numerical tests which all indicate that the proposed scheme can give very good approximations to the exact solution.

Keywords: two-phase flow, balance law, nonconservative, source term, numerical approximation, well-balanced scheme, Roe-type scheme, shock wave, rarefaction wave, contact discontinuity

MSC numbers: 35L65, 65M06, 76T10