报告人：Martin Stynes（北京计算科学研究中心）
邀请人：李东方
报告时间：2023年10月19日（星期四）16:00-18:00
报告地点：科技楼南楼702会议室
报告题目：Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations
报告摘要：The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^{\alpha}$ is the Caputo derivative of order $\alpha\in (0,1)$ and $\lambda, \gamma$ are positive parameters, is known to exhibit $O(t^{-\alpha/\gamma})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. We shall show that for the class of complete monotonicity-preserving schemes (which includes the L1 and Gr\"unwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-\alpha/\gamma})$ decay as $t_{n}\to\infty$. For the L1 scheme, this $O(t_{n}^{-\alpha/\gamma})$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.
报告人简介：Martin Stynes obtained his B.Sc and M.Sc. degrees from University College Cork, Ireland, then his PhD degree from Oregon State University, USA in 1977. After some other positions, he was at University College Cork from 1984 to 2012. Since 2013 he has been at Beijing CSRC, where he is a Chair Professor funded by the Chinese Government’s 1000 Talent Plan (Recruitment Program of Foreign Experts). He has worked for many years on the numerical solution of singularly perturbed differential equations; the book on this topic by Roos, Stynes and Tobiska is the standard international reference work (1st edition 1996, 2nd edition 2008). For the last 5 years he has worked mainly on fractional-derivative differential equations and their numerical solution. He is an editor of the journals Advances in Computational Mathematics, Applied Numerical Mathematics, and Computational Methods in Applied Mathematics.
