A meshless RBF method for computing a numerical solution of unsteady Burgers'-type equations

摘要

In this paper, we discuss a meshless method for the computation of a numerical solution of unsteady coupled Burgers'-type equations which are, in the most general case, nonlinear partial differential equations. Our approach is based on the interpolation of the solution by radial basis functions (RBFs) and is independent of the geometry of the domain. The existence and unicity of such of an interpolant will be established by the mean of linear algebra arguments. This spatial discretization transforms the initial problem into a system of ordinary differential equations (ODEs). We give a simple formulation of implicit Runge-Kutta (IRK) schemes that can be used to numerically solve this system of ODEs. The main difficulty in the implementation of an IRK method lies in the fact that it leads to a potentially large linear or nonlinear system that has to be solved at each time step. A matrix-GMRES approach, coupled with a Newton based linearization technique in the nonlinear case, will be used to numerically solve these systems. Some numerical examples will be provided to illustrate the performance of our approach.