报告人： 廖洪林 教授

报告题目： A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations

时间： 2026年10月29日（周四）10:00—12:00

地点： 西安交通大学数学与统计学院2-3会议室

报告摘要：

The rigorous stability analysis of high-order implicit-explicit linear multistep (IELM) methods for nonlinear parabolic equations by using discrete energy arguments is a longstanding open issue due to their non-A-stability property. A novel semi-generating function approach combined with a global discrete energy analysis is suggested for the stability and convergence of general IELM methods in solving nonlinear parabolic equations.

Inspired from the Grenander-Szego theorem for Toeplitz matrices, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IELM methods if the minimum eigenvalue of composite convolution kernels for the implicit part is properly large and the spectral norm bound of composite convolution kernels for the explicit part is properly small.

An indicator, called implicit-explicit controllability intensity, is then introduced to evaluate the degree of controllability of the implicit part over the explicit part. Some of the existing IELM methods, up to fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose an IELM method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model.

We also propose a new parameterized class of IELM methods, up to the ninth-order time accuracy, which satisfy the a priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing a proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.

报告人简介：

廖洪林，应用数学博士、教授、博士生导师，2018年至今任教于南京航空航天大学数学学院。2010年获理学博士学位，2001-2017年任教于原解放军理工大学、陆军工程大学。

学术研究方向为偏微分方程数值解，目前主要关注非线性相场以及多相流模型的高阶时间离散与时间自适应算法，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六十余篇。相关研究工作得到了国内外同行的广泛关注与认可、引发了很多后续研究工作，在WOS上他引3300余次。近年来有16篇研究论文入选ESI高被引论文，2023-2026连续四年入选Elsevier全球前2%顶尖科学家年度榜单、2025年入选Clarivate全球高被引研究者榜单。