Solving Eikonal equation in 2D and 3D by generalized finite difference method

Abstract

In this article we propose an implementation, for irregular cloud of points, of the meshless method called generalized finite difference method to solve the fully nonlinear Eikonal equation in 2D and 3D. We obtain the explicit formulas for derivatives and solve the system of nonlinear equations using the Newton–Raphson method to obtain the approximate numerical values of the function for the discretization of the domain. It is also shown that the approximation of the scheme used is of second order. Finally, we provide several examples of its application over irregular domains in order to test accuracy of the scheme, as well as comparison with order numerical methods.