Discrete gradient structure of the second-order integral averaged formula for nonlinear integro-diff

主讲人：廖洪林 南京航空航天大学教授

时间：2022年10月21日14：00

地点：腾讯会议 238 595 317

举办单位：数理学院

主讲人介绍：廖洪林，应用数学博士，2018年至今任教于南京航空航天大学数学学院。2001年在解放军理工大学获理学硕士学位，2010年在东南大学获理学博士学位，2001-2017年任教于解放军理工大学。学术研究方向为偏微分积分方程数值解，目前主要关注相场以及多相流模型的时间变步长离散与自适应算法, 在Math Comp，SIAM J Numer Anal, SIAM J Sci Comput，IMA J Numer Anal，J Comput Phys, Sci China Math等国内外专业期刊上发表学术研究论文三十余篇。

内容介绍：The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We bulid up a discrete gradient structure for a class of second-order variable-step approximations of fractional Riemann-Liouville integral and fractional Caputo derivative. Then certain variational energy dissipation laws at discrete levels of the corresponding variable-step Crank-Nicolson type methods are established for time-fractional Allen-Cahn and time-fractional Klein-Gordon type models. They are shown to be asymptotically compatible with the associated energy law of the classical Allen-Cahn and Klein-Gordon equations in the associated fractional order limits. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of our second-order methods.