Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives

Abstract

This article is devoted to the comparison between a high order difference scheme and radial basis functions (RBFs) meshless approach for the numerical solution of two-dimensional fractional Rayleigh–Stokes problem. The fractional derivative of problem is described in the Riemann–Liouville sense. In the high order difference scheme we discretize the spaces derivatives with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivatives to obtain a fully discrete implicit scheme. Also in the RBF meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the Kansa’s approach to approximate the spatial derivatives. We prove the stability and convergence for high order difference scheme using Fourier analysis and for time-discrete scheme in RBF method via energy approach. We compare the results of compact finite difference method and RBF meshless approach in terms of accuracy and CPU time. Also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method.