港澳六宝大全

讲座名称：Numerical Analysis of Spatially Homogeneous and Collisional Kinetic Equations: Error Estimates and Simulations

讲座人：何凌冰 教授

讲座时间：5月15日14:00-15:00

讲座地点：腾讯会议：532-935-163

讲座人介绍：

何凌冰教授现任清华大学数学科学系教授、应用数学与概率统计研究所所长。他长期致力于偏微分方程数学理论的前沿研究，围绕流体力学方程与动理学方程等领域开展了系统而深入的探索，在解的适定性、稳定性理论及高精度算法方面取得了若干具有重要影响力的成果。截至目前，他已在《Journal of the European Mathematical Society》（JEMS）、《Annales de l'ENS》、《Communications in Mathematical Physics》（CMP）及《Archive for Rational Mechanics and Analysis》（ARMA）等国际权威数学期刊上发表论文40余篇，研究成果获得了国内外同行的广泛认可，并多次受邀参加ICCM、Oberwolfach、Banff、CIRM等重要国际学术会议及研讨项目。

讲座内容：

This work develops and rigorously analyzes a unified Fourier spectral method for two fundamental collisional kinetic models: the homogeneous Landau equation and the spatially homogeneous cutoff Boltzmann equation. For both equations, we construct numerical solutions by truncating the velocity domain to a bounded box \( D_L \) and retaining \( N \) Fourier modes. A central contribution is the derivation of explicit error estimates for the proposed schemes, bounding the difference between the numerical and exact solutions in terms of the truncation parameters \( D_L \) and \( N \) for Maxwellian and hard potentials. Comprehensive numerical simulations are presented for both equations, which confirm the predicted convergence rates and demonstrate the methods' ability to capture essential solution dynamics, including the relaxation to equilibrium. Our results establish the first mathematically rigorous justification for Fourier spectral methods applied to the Landau equation while extending and complementing recent convergence analyses for the Boltzmann equation. Together, this work provides a complete, validated computational framework—from theoretical error analysis to practical implementation—for the reliable simulation of these foundational kinetic equations.

主办单位：数学与统计学院