Deterministic Solution of the Boltzmann Equation Using a Discontinuous Galerkin Velocity Discretization

We propose an approach for high order discretization of the Boltzmann equation in the velocity space using discontinuous Galerkin methods. Our approach employs a reformulation of the collision integral in the form of a bilinear operator with a time-independent kernel. In the fully non-linear case the complexity of the method is O(n(8)) operations per spatial cell where n is the number of degrees of freedom in one velocity direction. The new method is suitable for parallelization to a large number of processors. Techniques of automatic perturbation decomposition and linearisation are developed to achieve additional performance improvement. The number of operations per spatial cell in the linearised regime is O(n(6)). The method is applied to the solution of the spatially homogeneous relaxation problem. Mass momentum and energy is conserved to a good precision in the computed solutions.