Necessity of introducing non-integer shifted parameters by constructing high accuracy finite difference algorithms for a two-sided space-fractional advection–diffusion model

Abstract

In this study, we develop a second-order finite difference scheme based on the shifted convolution quadrature (SCQ) framework that approximates the space-fractional derivatives at a shifted node xn−θ where θ may not necessarily be an integer. By applying the proposed method for a space-fractional advection–diffusion equation in the spacial direction and the Crank–Nicolson scheme for the time variable discretization, we analyze the von Neumann stability for the fully discrete scheme. Further, we explore the impact of different θ on the robustness of our scheme for weak regular solutions and compare that with the shifted Grünwald–Letnikov formula. The results confirm the necessity of introducing non-integer shifted parameters θ.