学术报告

报告题目：Error analysis of a high-order numerical method on fitted meshes fora time-fractional diffusion problem

报告人：  陈虎博士  

报告时间：2019年1月9日下午3:30-4:30

报告地点：数统院307报告室

数学与统计学院

2019.1.8

摘要：In recent years, fractionalderivatives are used widely for modelling physical processes. Time-fractionaldiffusion equations are used to model abnormaldiffusion phenomena, where the mean square displacement is proportionalto tα with 0 < α < 1. There is much current interest in the constructionand analysis of numerical methods for the solution of such problems, whichtypically exhibit a weak singularity at the initial time t = 0. In [1] ahigh-order scheme for Caputo fractional derivatives of order α∈(0,1) is proposed andanalysed for time-fractional initial-value problems (IVPs) and initial-boundaryvalue problems (IBVPs), on temporal meshes that are fitted to the initial weaksingularity. In the IBVP the spatial domain is the unit square, where a spectralmethod is used, but other domains (in Rd for d ≥ 1) and other spatialdiscretisations (finite element, finite difference) could be handled bymodifying our analysis.  

It is proved in [1] that, when the fitted temporal mesh is chosensuitably, the scheme attains order 3 − α convergence in the discrete L∞ norm for the1-dimensional IVP, and second-order convergence in L∞(L2) for the IBVP.Numerical results demonstrate the sharpness of these theoretical convergenceestimates.

报告人简介：陈虎，北京计算科学研究中心博士后，师从外国千人计划MartinStynes教授， 2017年北京航空航天大学获得博士学位。在J. Comput. Phys.、J. Sci. Comput.、 J. Comput. Appl. Math.、 Comput. Math. Appl. 等国际期刊上以第一作者身份发表SCI论文7篇。已获得博士后科学基金资助1项，国家自然科学基金青年项目1项。研究方向为偏微分方程数值解，谱方法以及分数阶微分方程数值方法的理论分析。