A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation

Abstract: Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there is a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work towards that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [30] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed reduced basis method (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of: reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: 1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and 2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux.