RBF-FD method for the high dimensional time fractional convection-diffusion equation

Abstract

In this paper, a radial basis function finite difference (RBF-FD) method is proposed to solve the time fractional convection-diffusion equation in high dimensional space. Compact support Wendland RBF augmented with the polynomial terms is employed to determine a RBF-FD weights on neighboring nodes surrounding the center point for the spatial derivatives. And the time discretization is replaced by the shifted Grünwald formula. The method can lead to second-order convergence in space and time simultaneously. We get an efficient shape parameter by using an adaptive strategy to select the domain of the support. The main idea is to take into account the way the data points are distributed within the support domain in which the number of points is about constant. Moreover, the unconditional stability and convergence of the proposed method are deduced. Finally, numerical examples present that the proposed method leads to accurate results, fast evaluation, high efficiency for modeling and simulating for the fractional differential equations.