TY - JOUR
T1 - A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation
AU - Wang , Can
AU - Deng , Weihua
AU - Tang , Xiangong
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 739
EP - 771
PY - 2023
DA - 2023/11
SN - 20
DO - http://doi.org/10.4208/ijnam2023-1033
UR - https://global-sci.org/intro/article_detail/ijnam/22140.html
KW - Fractional diffusion equation, weak singularity, middle rectangle quadrature formula, modified $L1$ scheme, five-point difference scheme, graded mesh, $α$-robust.
AB - <p style="text-align: justify;">In this paper, we are concerned with the numerical solution for the two-dimensional
time fractional Fokker-Planck equation with the tempered fractional derivative of order $α.$ Although some of its variants are considered in many recent numerical analysis works, there are still
some significant differences. Here we first provide the regularity estimates of the solution. Then a
modified $L1$ scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t → 0^+,$ while the five-point difference
scheme is used in space. Stability and convergence are proved in the sense of $L^∞$ norm, getting a
sharp error estimate $\mathscr{O}(\tau^{{\rm min}\{2−α,rα\}})$ on graded meshes. Furthermore, the constant multipliers
in the analysis do not blow up as the order of Caputo fractional derivative $α$ approaches the
classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and
convergence orders of the presented schemes.</p>