讲座题目：Discontinuous Galerkin method for nonlinear Maxwell equations

主讲人： 吕茂辉 北京邮电大学 特聘副研究员

讲座时间： 2024年6月28日 (星期五)  14:00-15:30

讲座地点：清水河校区六号科研楼A344

主讲人简介： 吕茂辉, 北京邮电大学副研究员，硕士生导师。重庆大学理学博士学位，2018年-2019年受国家留学基金委资助访问美国伦斯勒理工学院数学科学系，2020年-2023年在中科院计算数学与科学工程计算研究所从事博士后工作。主持中国博士后科学基金特别资助项目、国家自然科学基金青年基金项目等。研究方向包括电磁波传播问题的高阶间断伽辽金方法和有限元方法、界面问题的高阶数值方法等。主要成果发表于Journal of Computational Physics, SIAM Journal on Applied Mathematics和Journal of Scientific Computing等期刊上。

讲座内容：In this talk, we consider the time domain Maxwell equations that augmented with a class of nonlinear constitutive polarization laws in high dimensions. The nontrivial discrete temporal treatment of the nonlinearity in the ordinary differential equations that encode the Kerr and Raman effects (Bokil et al. 2017), is first generalized to higher spatial dimensions. To further improve the computational efficiency in dealing with the nonlinearity, we apply nodal DG methods in space. Energy stability is proved for the semi-discrete in time and in space schemes as well as for the fully-discrete schemes. Under some assumptions on the strength of nonlinearity, error estimates are established for the semi-discrete in space methods, and, in particular, optimal accuracy is achieved for the methods on Cartesian meshes with Qk-type elements and alternating fluxes. Attention is paid to the role of the nodal form of the DG discretizations in the analysis. We will also present a second order linear scheme for the nonlinear Maxwell equations. Numerical examples are presented to validate the accuracy, energy stability, and computational efficiency of the proposed schemes. We further illustrate the performance of the methods through physically relevant experiments involving spatial soliton propagation and airhole scattering in realistic glasses.