Collective Dynamics in Biological and Social Systems

Asymptotic-preserving stochastic Galerkin schemes for the Boltzmann equation with uncertainty

Jingwei Hu

Purdue University


We develop a stochastic Galerkin method for the nonlinear Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the Fourier-spectral method (in velocity space) allows one to compute the collision operator efficiently. When the Knudsen number is small, we propose a new technique to overcome the stiffness induced by collision operator. The resulting scheme is uniformly stable in both kinetic and fluid regimes, which offers a possibility of solving the compressible Euler equations with random inputs.