A Riemann solver and upwind methods for a two-phase flow model in non-conservative form

We present a theoretical solution for the Riemann problem for the five-equation two-phase non-conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non-conservative methods to solve the general initial-boundary value problem for the two-phase flow model in non-conservative form. The basic upwind scheme constructed is the non-conservative analogue of the Godunov first-order upwind method. Second-order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions.