A non‐standard finite difference method for space fractional advection–diffusion equation

Abstract

AbstractIn this paper, a non‐standard finite difference scheme is developed to solve the space fractional advection–diffusion equation. By using Fourier–Von Neumann method, we prove that non‐standard finite difference scheme is unconditionally stable. We further discuss the convergence of numerical method and give the order of convergence. The numerical examples show that the non‐standard finite difference method can effectively reduce the maximum error and improve the accuracy of numerical solution in contrast to classical numerical methods. Moreover, we find that our numerical scheme is very flexible, when we optimize the denominator function of time and space simultaneously, the performance is the best. These studies show that the non‐standard finite difference scheme is feasible and efficient for solving fractional partial differential equations.