GFMxP: A novel Ghost Fluid Method to solve the variable coefficient Poisson equation

The solution of a variable coefficient Poisson equation, where the same coefficient, the unknown function and its spatial derivatives may be discontinuous, is an interesting and challenging problem, whose solution plays a fundamental role in the simulation of many physical phenomena and, in particular, of an incompressible multiphase flow. On the other hand, the Ghost Fluid Method (GFM) represents an effective numerical technique to manage derivative terms in presence of an interface (that is, a discontinuity) and to preserve its own sharp nature. Then, the method is a natural candidate to deal with the above mentioned problem. The equation, however, may be characterized by a significant jump of the coefficient across the interface, a condition which still leaves room for open questions and in which the original GFM-based approach seems not to work properly. The paper provides a comprehensive answer to these questions and proposes a novel, simpler implementation of the method (named GFMxP), testing its efficiency and robustness by a lot of well-known numerical examples.