The GFMxP and the basic extrapolation of the ghost values to solve the Poisson equation for discontinuous functions

In a recent paper, a novel coding of the Ghost Fluid Method for the variable coefficient Poisson equation with discontinuous functions (named GFMxP) was proposed. A lot of numerical tests, with all the required quantities available in a analytic form, were used to demonstrate the ability of the new procedure in modeling a sharp interface and to check the accuracy order of the solutions. In practical applications, however, the real difficulty stands in the estimation of the so-called “ghost values”, that is the values at points where the function is not only unknown, but even not defined. These values allow to compute the corrective terms enabling the use of standard finite difference formulas in presence of a singularity and/or a discontinuity, and can be only determined through some extrapolation procedure, whose truthfulness is essential to achieve a reliable result. The paper deals with such a basic issue, by testing different numerical strategies and demonstrating the strict relationship between the order of the adopted fit-model, the order of the solving scheme for the Poisson equation and the accuracy of the final solution.