  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 applied to treat kinetic
equations, including the Vlasov--Poisson and the Boltzmann equation, where 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 investigate the error analysis for a dynamical low-rank
algorithm applied to the multi-scale 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 one 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.
