A Deterministic-Stochastic Approach to Compute the Boltzmann Collision Integral in O(MN) Operations

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision operator with O(MN) operations, where N is the number of the discrete velocity points and M < N. The approach is formulated using a bilinear convolution form of the Galerkin projection of the collision operator and discontinuous Galerkin (DG) discretizations of the collision operator. Key ingredients of the new approach are singular value decomposition (SVD) compression of the collision kernel and approximations of the solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. The developed method is significantly faster than the full deterministic DG velocity discretization of the collision integral. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.