A conservative reduced collision model for the Boltzmann equation

Abstract

The evaluation of the quadratic collision operator is a major computational bottleneck in solving the Boltzmann equation owing to the high dimensionality and the nonlinear structure of the collision term. In this work, we develop a conservative reduced collision (RC) model within the discrete velocity framework. The bilinear collision operator is approximated through a kernel tensor preassembled using problem-dependent reduced bases for the distribution function and the collision term. Consequently, online evaluation of the quadratic collision operator is reduced to tensor contractions. Exact discrete conservation of mass, momentum, and energy is enforced by constructing the collision basis in the orthogonal complement of the space spanned by the discrete collision invariants. The resulting model can be incorporated into existing deterministic Boltzmann solvers. Numerical experiments on the spatially homogeneous relaxation, the steady channel and lid-driven cavity flows, and an unsteady diffusion problem show that the proposed RC model attains the accuracy of the fast Fourier spectral method (FSM) at moderate ranks. Since the kernel tensor is assembled once in the offline stage, the online cost of each solve is reduced by orders of magnitude, so that the total cost is substantially lower even when the offline stage is included.