Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Despite its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [Trans.Amer.Math.Soc. 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful L2 estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This is joint work with Kunlun Qi and Tong Yang.

Jingwei Hu is an associate professor of mathematics at Purdue University, where she also holds a courtesy appointment in School of Aeronautics and Astronautics. Before moving to Purdue, she is a postdoc fellow at The university of Texas at Austin. Prof. Hu received her Ph.d in Mathematics from University of Wisconsin-Madison in 2011, and B.S. in Mathematics, from Peking University in 2006. Her research is mainly focused on development and analysis of efficient and structure-preserving numerical methods for kinetic equations and related problems arising in multiscale modeling and simulation.