Effective numerical treatment of sub-diffusion equation with non-smooth solution

TL;DR

This paper introduces a high-order numerical scheme for sub-diffusion equations with non-smooth solutions, combining transformation, smooth operators, and spectral methods to achieve high convergence rates in simulating anomalous diffusion.

Contribution

The paper develops a novel high-order numerical method that effectively handles low-regularity solutions in sub-diffusion equations using transformation and spectral techniques.

Findings

High convergence rate achieved despite non-smooth solutions

Method verified through numerical experiments demonstrating efficiency

Applicable to real physical anomalous diffusion scenarios

Abstract

In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nystrom method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.