Numerical analysis for subdiffusion problem with non-positive memory

TL;DR

This paper develops a numerical scheme for subdiffusion problems with non-positive memory, proving stability and accuracy, and extends existing analysis to more complex subdiffusive models with variable exponents.

Contribution

It introduces a novel numerical method handling non-positive memory kernels in subdiffusion equations, with rigorous stability and accuracy analysis.

Findings

The scheme is stable under non-positive memory conditions.

First-order temporal accuracy is achieved.

Numerical experiments confirm theoretical results.

Abstract

This work considers the subdiffusion problem with non-positive memory, which not only arises from physical laws with memory, but could be transformed from sophisticated models such as subdiffusion or subdiffusive Fokker-Planck equation with variable exponent. We apply the non-uniform L1 formula and interpolation quadrature to discretize the fractional derivative and the memory term, respectively, and then adopt the complementary discrete convolution kernel approach to prove the stability and first-order temporal accuracy of the scheme. The main difficulty in numerical analysis lies in the non-positivity of the kernel and its coupling with the complementary discrete convolution kernel (such that different model exponents are also coupled), and the results extend those in [Chen, Thom\'ee and Wahlbin, Math. Comp. 1992] to the subdiffusive case. Numerical experiments are performed to substantiate the theoretical results.