A DETERMINISTIC-STOCHASTIC METHOD FOR COMPUTING THE BOLTZMANN COLLISION INTEGRAL IN O(MN) OPERATIONS

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with O(MN) operations, where N is the number of the discrete velocity points and M < N. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.