Fast evaluation of the Boltzmann collision operator using data driven reduced order models

Abstract

We consider application of reduced order models (ROMs) to accelerating solutions of the spatially homogeneous Boltzmann equation for the class of problems of spatially homogeneous relaxation of sums of two homogeneous Gaussian densities. Approximation spaces for the ROMs are constructed by performing singular value decomposition of the solution data matrix and extracting principal singular vectors/modes. The first ROM results from a straightforward Galerkin discretization of the spatially homogeneous Boltzmann equation using a truncated basis of the singular vectors. The model approximates solutions to the Boltzmann equation accurately during early stages of evolution. However, it suffers from presence of ROM residuals at later stages and exhibits slowly growing modes for larger ROM sizes. In order to achieve stability, the second ROM evolves the difference between the solution and the steady state. The truncated singular vectors are orthogonalized to the steady state and modified locally to enforce zero density, momentum, and temperature moments. Exponential damping of ROM residuals is introduced to enforce physical accuracy of the steady state solution. Solutions obtained by the second ROM are asymptotically stable in time and provide accurate approximations to solutions of the Boltzmann equation. Complexity for both models is O(K3) where K is the number of singular vectors retained in the ROMs. For the considered class of problems, the models result in up to three orders of magnitude reduction in computational time as compared to the O(M2) nodal discontinuous Galerkin (DG) discretization, where M is the total number of velocity points.