Modified BDF2 schemes for subdiffusion models with a singular source term

TL;DR

This paper introduces a modified BDF2 time stepping scheme for subdiffusion equations with singular source terms, achieving second-order accuracy despite the source term's irregularity.

Contribution

A novel time stepping scheme using regularization and a modified BDF2 convolution quadrature that maintains second-order accuracy for nonsmooth source terms.

Findings

Proven second-order convergence of the scheme.

Numerical results confirm theoretical accuracy.

Effective handling of weakly singular source terms.

Abstract

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using a $k$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results.