Jingwei Hu is an Assistant Professor of Mathematics at Purdue University. She received her B.S. in mathematics from Peking University in 2006, and Ph.D. from University of Wisconsin-Madison in 2011. Before joining Purdue in 2014, she was a postdoctoral fellow at the Institute for Computational Engineering and Sciences (ICES), the University of Texas at Austin. Her research interests include the design and analysis of efficient numerical methods and fast algorithms for problems arising in kinetic theory, wave propagations, seismic imaging, etc.

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernel. Compared to the classical spectral method (Pareschi and Russo, 2000) which requires $O(N^6)$ memory to store precomputed weights and $O(N^6)$ numerical complexity ($N$ is the number of discretization points in each velocity dimension), the new method runs in complexity $O(MN^4\log N)$ ($M$ is the total discrete points on a sphere and $M\ll N^2$). Furthermore, it requires no precomputation for variable hard sphere (VHS) model, and only $O(MN^4)$ memory to store precomputed function for general collision kernels. Although a faster spectral method exists (Mouhot and Pareschi 2006) (complexity $O(MN^3\log N)$), it only works for hard sphere molecule which limits its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.