Solving second order non-linear hyperbolic 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 hyperbolic PDEs. This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the PDEs are based on the existence of a positive definite matrix which is obtained using moving least squares approximation and Taylor series development. Consistency and stability are shown in this paper for semilinear and quasilinear hyperbolic equations. This paper shows the application of the GFDM for solving second order non-linear hyperbolic problems.