Third-order time stepping methods for superdiffusion using weighted and shifted Grünwald–Letnikov formulae with nonsmooth data

Abstract

In this paper, we study a numerical method for the Caputo time fractional wave equation with nonsmooth data. We first introduce a class of third-order approximations, known as weighted and shifted Grünwald-Letnikov approximations, to approximate the Caputo fractional derivative. Based on this, we develop a new time stepping method for solving the time fractional wave equation. After applying corrections to several initial steps, the proposed time stepping method achieves a convergence order of O(k3)$$O(k^3)$$ for nonsmooth data, where k denotes the time step size. We also analyze the stability regions of the proposed time stepping method, which show that the scheme is unconditionally stable for α∈(1,1.94)$$ \alpha \in (1, 1.94) $$, and conditionally stable for α∈[1.94,2)$$ \alpha \in [1.94, 2) $$. Numerical experiments are presented to validate the theoretical findings.