A numerical method for two-phase flow consisting of separate compressible and incompressible regions

We propose a numerical method for modeling two phase flow consisting of separate compressible and incompressible regions. This is of interest, for example, when numerically modeling the combustion of fuel droplets or the shock induced mixing of liquids. We use the level set method to track the interface between the compressible and incompressible regions, as well as the Ghost Fluid Method (GFM) to create accurate discretizations across the interface. The GFM is particularly effective here since the equations differ in both number and type across the interface. Further, we use multigrid methods and also the Preconditioned Conjugate Gradient (PCG) method with an Incomplete Choleski preconditioner to solve the Poisson equation that arises in these computations. We compare the performance of these two algorithms. The numerical method is presented in two spatial dimensions with numerical examples in both one and two spatial dimensions, while three dimensional extensions are straightforward.