A second-order difference scheme for the time fractional substantial diffusion equation
This work provides a more accurate numerical method for solving fractional diffusion equations with substantial derivatives, which is relevant for researchers in applied mathematics and computational physics.
The authors developed a second-order approximation for the fractional substantial derivative and applied it to a time-fractional substantial diffusion equation, achieving a fully discrete compact difference scheme with proven stability and convergence. Numerical examples demonstrate the scheme's reliability and efficiency.
In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald 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 Grünwald-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.