A second-order difference scheme for the time fractional substantial diffusion equation

TL;DR

This paper develops a second-order numerical scheme for solving the time fractional substantial diffusion equation, combining a modified Gr"{u}nwald formula with a compact spatial discretization, and demonstrates its stability, convergence, and effectiveness through numerical tests.

Contribution

It introduces a novel second-order approximation for the fractional substantial derivative and integrates it into a fully discrete scheme with proven stability and convergence.

Findings

Scheme is stable and convergent under smooth conditions.

Numerical examples confirm the scheme's accuracy and efficiency.

Improved algorithm handles nonsmooth solutions effectively.

Abstract

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Gr\"{u}nwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully discrete Gr\"{u}nwald-Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme.