报告题目： A unified framework on the original energy laws of three effective classes of Runge-Kutta methods for phase field crystal type models

报 告 人：廖洪林 教授 （南京航天航空大学）

邀 请 人：宋怀玲

报告时间：2025/12/19 周五 10:00-11:30

报告地点：数学院425

摘    要：

The main theoretical obstacle to establishing the original energy dissipation laws of Runge-Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge-Kutta methods, including the additive implicit-explicit Runge-Kutta, explicit exponential Runge-Kutta, and corrected integrating factor Runge-Kutta methods, for the Swift--Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge--Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge-Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge-Kutta methods for dissipative semilinear parabolic problems.

报告人简介：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。2010年在东南大学获理学博士学位，2001-2017年任教于原解放军理工大学。长期从事偏微分方程数值解的学术研究工作，侧重于非线性相场、不可压Navier-Stokes模型和多相流耦合系统高阶自适应算法的设计与理论分析；构造了离散互补卷积核、离散正交卷积核、平均耗散速率等离散分析工具，在非均匀以及高精度时间离散方法方面形成了一套独具特色的算法设计与理论分析方法。已在SIAM Journal on Numerical Analysis、Mathematics of Computation、SIAM Journal on Scientific Computing、Journal of Computational Physics、IMA  Journal of Numerical Analysis、Science China Mathematics等国内外计算数学高水平专业期刊上发表学术论文六十余篇；相关研究工作也得到了国内外同行的广泛关注与认可，其中ESI高被引论文15篇，2023-2025年连续入选全球前2%顶尖科学家年度榜单（计算数学），2025年入选Clarivate全球高被引学者榜单。