A stable RBF-FD method for solving two-dimensional variable-order time fractional advection-diffusion equation

Abstract

The meshless method is used in this paper to provide a numerical solution for a two-dimensional variable-order time fractional advection-diffusion equation. The radial basis function and the finite difference scheme on two-dimensional (2-D) arbitrary domains are the foundations of this accurate and reliable meshless approach. The name of this approach is radial basis function-generated finite differences (RBF-FD). The finite difference technique and the RBF-FD approximation are both employed for discrete time and space domains, respectively. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. The advantage of the RBF-FD method is that it has high-order convergence rates and its ability to manage sparse node layouts in arbitrary domains. The use of polyharmonic splines (PHS) as a RBF and the absence of the shape parameter are also a positive points for the mentioned method. The effectiveness and precision of the proposed method are examined with three distinct examples, incorporating two model examples and a real-world application of contamination transportation phenomena.