A sharp $\alpha$-robust $L1$ scheme on graded meshes for two-dimensional time tempered fractional Fokker-Planck equation

TL;DR

This paper develops a robust numerical scheme for solving two-dimensional time tempered fractional Fokker-Planck equations, achieving high accuracy and stability on graded meshes, and verifies its effectiveness through numerical experiments.

Contribution

It introduces a sharp $ ext{α}$-robust $L1$ scheme on graded meshes that handles solution singularities and maintains stability as $ ext{α}$ approaches 1, with proven convergence and error estimates.

Findings

The scheme achieves the error order $ ext{O}( au^{ ext{min}\{2- ext{α}, r ext{α} ight\")

The constant multipliers do not blow up as $ ext{α}$ approaches 1.

Numerical experiments confirm the scheme's effectiveness and convergence order.

Abstract

In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $\alpha$. Although some of its variants are considered in many recent numerical analysis papers, there are still some significant differences. Here we first provide the regularity estimates of the solution. And then a modified $L$1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t\rightarrow 0^{+}$, while the five-point difference scheme is used in space. Stability and convergence are proved in the sence of $L^{\infty}$ norm, then a sharp error estimate $\mathscr{O}(\tau^{\min\{2-\alpha, r\alpha\}})$ is derived on graded meshes. Furthermore, unlike the bounds proved in the previous works, the constant multipliers in our analysis do not blow up as the Caputo fractional derivative $\alpha$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence order of the presented algorithms.