学术报告20251208：Numerical analysis of two-dimensional time fractional Allen-Cahn equation under new non-uniform mesh construction strategy

报告人：丁恒飞 教授 广西师范大学

报告题目：Numerical analysis of two-dimensional time fractional Allen-Cahn equation under new non-uniform mesh construction strategy

摘 要：In this report, we introduce a new non-uniform mesh construction strategy for the two-dimensional time-fractional Allen-Cahn equation. In order to facilitate the construction of efficient numerical algorithms, we first transform the original equation into an equivalent integral equation. Because the solution of this equation has weak singularity at the initial time, one of the most effective ways to deal with it is to use numerical integration formulas with non-uniform meshes to approximate the fractional integral operator contained in it. Therefore, we have constructed a new non-uniform mesh that is superior to the strategies in existing literatures, and combined the fractional rectangle method and the fractional trapezoid method to obtain two new numerical integration formulas for approximating the fractional integral. Next, by applying the central difference formula to the spatial derivative, two efficient numerical schemes for solving the equivalent integral equation can be obtained. Furthermore, we conduct a detailed study of the discrete maximum principle and energy decay under certain conditions for the fractional trapezoidal numerical scheme. Meanwhile, the stability and convergence have also been rigorously proven, and the results show that its time and spatial convergence orders are both second-order. Finally, we provide a numerical example that demonstrates the superiority of our constructed non-uniform mesh over existing meshes.

报告人简介：丁恒飞教授（三级）、博/硕士生导师（学术型）、广西壮族自治区高层次人才（D类）、广西师范大学漓江学者（A类）、广西数学学会常务理事、桂林市数学学会常务理事，广西科协智库决策咨询专家库专家。目前主要从事整数和分数阶微分方程的建模、分数阶微分方程定性理论分析、分数阶导数高阶数值逼近和分数阶微分方程高阶数值算法及其应用研究。在国际知名期刊Journal of Computational Physics、Journal of Scientific Computing、Physics of Fluids、IMA Journal of Applied Mathematics和Applied Numerical Mathematics等发表学术论文40余篇。主持国家自然科学基金项目3项（在研1项，结题2项）、省自然科学基金项目2项（均结题），参与项目若干。现任SCI二区Top期刊《Mathematics and Computers in Simulation》和《中国理论数学前沿》学术编委。

报告时间：2025年12月8日，14:15--17:15

报告地点：腾讯会议（817-194-057）

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内蒙古大学数学科学学院

2025年12月3日