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

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.

Citation

He, Y., Chen, J., Yan, Y., Chen, X., Liu, X., & Ding, P. (2026). Third-order time stepping methods for superdiffusion using weighted and shifted Grünwald–Letnikov formulae with nonsmooth data. Journal of Scientific Computing, 106, article number 4. https://doi.org/10.1007/s10915-025-03088-5

Publisher

Springer

Journal

Journal of Scientific Computing

URI

http://hdl.handle.net/10034/629808

DOI

10.1007/s10915-025-03088-5

Type

Article

Language

en

Description

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10915-025-03088-5

ISSN

0885-7474

EISSN

1573-7691

Additional Links

https://link.springer.com/article/10.1007/s10915-025-03088-5