New numerical approaches to multiphase flow modelling

We present new numerical approaches for solving systems of partial differential equations associated with mathematical models for multiphase flows. We are concerned with the construction of modern numerical methods for solving the equations for hyperbolic models in conservative or non-conservative form. Here, we apply new approximate Riemann solvers for two-phase flow, whereby, a closed-form non-iterative solution can be obtained,1 and a new approach for general hyperbolic systems called EVILIN.2 In order to produce upwind numerical methods, the local approximate Riemann solution provides the necessary information to compute numerical fluxes that can be used in the finite volume approach or the Discontinuous Galerkin approach. We utilize these approximate Riemann solvers locally to produce upwind numerical methods in the finite volume framework suitably modified to deal with systems in non-conservative form. Non-oscillatory schemes of second-order accuracy are then designed following the TVD approach. In addition, we construct second-order numerical schemes for multiphase flows following the recently proposed ADER3 approach, which also permits the handling of source terms to a high order of accuracy. We perform a comprehensive and systematic assessment of the numerical methods constructed using reference numerical solutions and exact solutions that we have obtained for special cases.