Dynamical Low-Rank Integrator for the Linear Boltzmann Equation: Error Analysis in the Diffusion Limit
Zhiyan Ding, Lukas Einkemmer, Qin Li
Abstract
Dynamical low-rank algorithms are a class of numerical methods that compute low-rank approximations of dynamical systems. This is accomplished by projecting the dynamics onto a low-dimensional manifold and writing the solution directly in terms of the low-rank factors. The approach has been successfully applied to many types of differential equations. Recently, efficient dynamical low-rank algorithms have been proposed in [L. Einkemmer, A Low-Rank Algorithm for Weakly Compressible Flow, arXiv:1804.04561, 2018; L. Einkemmer and C. Lubich, SIAM J. Sci. Comput., 40 (2018), pp. B1330--B1360] to treat kinetic equations, including the Vlasov--Poisson and the Boltzmann equation. There it was demonstrated that the methods are able to capture the low-rank structure of the solution and significantly reduce numerical cost, while often maintaining high accuracy. However, no numerical analysis is currently available. In this paper, we perform an error analysis for a dynamical low-rank algorithm applied to the multiscale linear Boltzmann equation (a classical model in kinetic theory) to showcase the validity of the application of dynamical low-rank algorithms to kinetic theory. The equation, in its parabolic regime, is known to be rank 1 theoretically, and we will prove that the scheme can dynamically and automatically capture this low-rank structure. This work thus serves as the first mathematical error analysis for a dynamical low-rank approximation applied to a kinetic problem.