Error analysis of an asymptotic preserving dynamical low-rank integrator for the multi-scale radiative transfer equation
share
Dynamical low-rank algorithm 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 the numerical effort, while often maintaining good accuracy. However, no numerical analysis is currently available. In this paper, we perform an error analysis for a dynamical low-rank algorithm applied to a classical model in kinetic theory, namely the radiative transfer equation. The model used here includes a small parameter, the Knudsen number. This setting is particularly interesting since the solution is known to be rank one in certain regimes. We will prove that the scheme dynamically and automatically captures the low-rank structure of the solution, and preserves the fluid limit on the numerical level. This work thus serves as the first mathematical error analysis for a dynamical low rank approximation applied to a kinetic problem.
READ FULL TEXT