报告题目一：Hermite-Galerkin spectral method for Klein-Gordon-Schrodinger system on unbounded domains: Conservation of invariants

报 告 人：郭士民（西安交通大学）

报告时间：2024年12月13日9:30—10:30

报告地点：腾讯会议 691203563

报告摘要：In this talk, we shall consider the Hermite-Galerkin spectral method for the Klein-GordonSchrodinger (KGS) system. First, we construct the finite difference/spectral method for the ddimensional KGS system to conserve three of the most important invariants, namely, mass, energy, and momentum. Regarding the mass and momentum conservation laws as d+1 globally physical constraints, we carefully combine the exponential scalar auxiliary variable (ESAV) approach and the Lagrange multiplier approach to construct the ESAV-Lagrange multiplier reformulation of the KGS system, thereby preserving its original energy conservation law. Secondly, for the nonlocalin-space KGS system in multi-dimensional unbounded domains, we use the Hermite-Galerkin spectral method with a scaling factor for spatial approximation and the Crank-Nicolson scheme for temporal discretization, which conserves the nonlocal energy at the fully discrete level.

报告人简介：郭士民，西安交通大学教授、博士生导师，主要研究方向为高精度数值算法、计算等离子体 物理；在 SIAM Journal on Scientific Computing、Journal of Computational Physics 等期刊上发 表多篇学术论文，主持国家自然科学基金面上项目、国家重点研发计划子课题等多项科研项 目；博士学位论文入选“2016 年度陕西省优秀博士学位论文”，荣获 2019 年度陕西省自然科 学奖二等奖。

报告题目二：An Improved Local Min-Orthogonal Method for Finding Multiple Saddle Solutions to Semilinear PDEs

报 告 人：陈先进（中国科学技术大学）

报告时间：2024年12月13日10:30—11:30

报告地点：腾讯会议 691203563

报告摘要：Local Min-Orthogonal (LMO) method is an efficient numerical method to solve nonlinear elliptic equations or systems for their multiple saddle-solutions. With a given finite-dimensional subspace L, the LMO method can find multiple saddle solutions outside such L. In this talk, the L-⊥ selection, the separation condition and the continuity condition used in the framework of the LMO method are successively improved or weakened so that they are not only closer to the real algorithm’s implementation but also able to improve its convergence analysis. A new step- size rule and a new local characterization on saddles are then established, based on which an improved LMO (also called LMO+) method is developed. The new method can overcome some limitations of the LMOtype method and enjoy some important advantages, such as boundedness of the iterated sequence and of its corresponding energy functional. In the end, numerical examples are presented to demonstrate the effectiveness of the new method.

报告人简介：陈先进，德克萨斯 A&M 大学（美）数学博士，明尼苏达大学应用数学研究所博士后，现任 教于中国科学技术大学。目前主要从事非线性偏微分方程(组)不稳定多解的分析与计算方面 的研究，并在该领域取得了一些原创性的研究成果，成果发表在 Math. Comp., J. Sci. Comput., Appl. Numer. Math., Physica D, J. Comput. Appl. Math., Comm. in Math. Sci., Numer. Meth. PDEs 等知名期刊上