A Conservative Discontinuous Galerkin Scheme with O(N ) Operations in Computing Boltzmann Collision Weight Matrix

In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of “shifting symmetry” in the weight matrix to reduce the computing complexity from theoretical O(N) down to O(N), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpenMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.