Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM)

Abstract

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): 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 parabolic PDEs. This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Criteria for convergence of fully explicit method using GFDM for different non linear parabolic PDEs are given. 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.