The extended Ghost Fluid Method (xGFM): recovering convergence of the fluxes when solving Poisson equations with jump conditions.

Among the various numerical methods that have been developed over the last decades to address the Poisson problem with jump conditions across an irregular interface, the "Boundary Condition Capturing Method" by Liu, et al., 2000 (commonly referred to as the "Ghost Fluid Method") has established itself as a first-choice tool in (sharp) multiphase flow simulations. Its small discretization stencil, its ease of implementation and the symmetric positive-definiteness of its linear system of equations make the method very robust and thus highly suited for large-scale simulations. Nevertheless, the GFM's weakness lies in the lack of convergence for the gradients of the solution (and thus the flux vectors) which may pose serious accuracy issues, especially in the context of projection methods.

In this presentation, we show a rather simple fix to recover convergence for the gradient of the solution. The technique does not alter the discretization stencil nor the symmetric positive-definiteness of the linear system of equations. Illustrations (including multiphase-flow simulations) are shown.