Solving second order non-linear elliptic partial differential equations using generalized finite difference method

Abstract

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (pde’s): wave propagation, advection–diffusion, plates, beams, etc.The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear elliptic pde’s.This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based on the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Also it is shown that in 2D a regular neighbourhood of eight nodes can be regarded as a generalization of a classical finite difference formula with a sixth order truncation error.This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.