A Nonlinear Projection-Based Iteration Scheme with Cycles over Multiple Time Steps for Solving Thermal Radiative Transfer Problems

In this paper we present a multilevel projection-based iterative scheme for solving thermal radiative transfer problems that performs iteration cycles on the high-order Boltzmann transport equation (BTE) and low-order moment equations. Fully implicit temporal discretization based on the backward Euler time-integration method is used for all equations. The multilevel iterative scheme is designed to perform iteration cycles over collections of multiple time steps, each of which can be interpreted as a coarse time interval with a subgrid of time steps. This treatment is demonstrated to transform implicit temporal integrators to diagonally-implicit multi-step schemes on the coarse time grid formed with the amalgamated time intervals. A multilevel set of moment equations are formulated by the nonlinear projective approach. The Eddington tensor defined with the BTE solution provides exact closure for the moment equations. During each iteration, a number of chronological time steps are solved with the BTE alone, after which the same collection of time steps is solved with the moment equations and material energy balance. Numerical results are presented to demonstrate the effectiveness of this iterative scheme for simulating evolving radiation and heat waves in 2D geometry.