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Xiangong Tang

Publications and source records attributed to Xiangong Tang.

2 recordsLinked to original sources
arXiv2022

Numerical method for the Fokker-Planck equation of Brownian motion subordinated by inverse tempered stable subordinator with drift

In this work, based on the complete Bernstein function, we propose a generalized regularity analysis including maximal $\mathrm{L}^p$ regularity for the Fokker--Planck equation, which governs the subordinated Brownian motion with the inverse tempered stable subordinator that has a drift. We derive a generalized time--stepping finite element scheme based on the backward Euler convolution quadrature, and the optimal-order convergence of the numerical solutions is established using the proven solution regularity. Further, the analysis is generalized to more general diffusion equations. Numerical experiments are provided to support the theoretical results.

math.NA↗
arXiv2022

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

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 $α$. 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 $\mathscr{O}(τ^{\min\{2-α, rα\}})$ 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 $α$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence order of the presented algorithms.

math.NA↗