A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows (Communications in Computational Physics, 2011)
In this paper, a new sharp-interface approach to simulate compressiblemultiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservativediscontinuous Galerkin finite element scheme to evolve an indicator functionthat tracks the material interface. At the interface our method applies ghost cellsto compute the numerical flux, as the ghost fluid method. However, unlike the originalghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derivedfrom an approximate-state Riemann solver, similar to the approach proposed in [25],but based on a much simpler formulation. Our formulation leads only to one singlescalar nonlinear algebraic equation that has to be solved at the interface, instead ofthe system used in [25]. Away from the interface, we use the new general Osher-typeflux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemannsolver, applicable to general hyperbolic conservation laws. The time integrationis performed using a fully-discrete one-step scheme, based on the approaches recentlyproposed in [5, 7]. This allows us to evolve the system also with time-accurate localtime stepping. Due to the sub-cell resolution and the subsequent more restrictivetime-step constraint of the DG scheme, a local evolution for the indicator function isapplied, which is matched with the finite volume scheme for the solution of the Eulerequations that runs with a larger time step. The use of a locally optimal time stepavoids the introduction of excessive numerical diffusion in the finite volume scheme.Two different fluids have been used, namely an ideal gas and a weakly compressiblefluid modeled by the Tait equation. Several tests have been computed to assess theaccuracy and the performance of the new high order scheme. A verification of ouralgorithm has been carefully carried out using exact solutions as well as a comparisonwith other numerical reference solutions. The material interface is resolved sharplyand accurately without spurious oscillations in the pressure field.