On the boundary conditions for GFMxP high-order schemes on staggered grids in the simulation of incompressible multiphase flows

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The simulation of incompressible multiphase flows through the so-called fractional step method needs to solve a variable coefficient Poisson equation for discontinuous functions. Recently, it has been shown how the solution of this equation may be found out through a novel coding of the Ghost Fluid Method (named GFMxP), by avoiding any fit to evaluate the interface position and providing, anyhow, a perfect sharp modeling of the same interface. Furthermore, the accuracy order of the numerical solutions exactly corresponds to the order of the adopted finite difference scheme. The effectiveness and reliability of the new procedure were successfully checked by a lot of tests. However, the a-priori knowledge of the unknown function allowed to elude a fundamental aspect of the numerical approach: the appropriate encoding of the boundary conditions. This topic has often been debated in the past, especially from a theoretical viewpoint, and still represents a rather thorny point in the whole simulation process. The paper shows how to handle the problem in practice and in the context of the GFMxP approach, i.e. by accounting for the presence of the discontinuity and the possible use of high-order solving schemes on a staggered grid.