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Dynamical low-rank algorithms are a class of numerical methods that computelow-rank approximations of dynamical systems. This is accomplished byprojecting the dynamics onto a low-dimensional manifold and writing thesolution directly in terms of the low-rank factors. The approach has beensuccessfully applied to many types of differential equations. Recently,efficient dynamical low-rank algorithms have been applied to treat kineticequations, including the Vlasov--Poisson and the Boltzmann equation, where itwas demonstrated that the methods are able to capture the low-rank structure ofthe solution and significantly reduce numerical cost, while often maintaininghigh accuracy. However, no numerical analysis is currently available.  In this paper, we investigate the error analysis for a dynamical low-rankalgorithm applied to the multi-scale linear Boltzmann equation (a classicalmodel in kinetic theory) to showcase the validity of the application ofdynamical low-rank algorithms to kinetic theory. The equation, in its parabolicregime, is known to be rank one theoretically, and we will prove that thescheme can dynamically and automatically capture this low-rank structure. Thiswork thus serves as the first mathematical error analysis for a dynamicallow-rank approximation applied to a kinetic problem.

Dynamical low-rank integrator for the linear Boltzmann equation: error analysis in the diffusion limit