A Staggered Lattice Boltzmann Method for the Radiative Transfer Equation

Abstract

This paper introduces a method for the numerical approximation of solutions of the mono-kinetic Radiative Transfer Equation, adapting some of the Lattice Boltzmann Method features. The main difference between the Radiative Transfer Equation and the Boltzmann Equation, used in the classical Lattice Boltzmann Method framework, lies in the constrained norm of the velocity field appearing in the advection operator. This small difference leads to off-grid propagation if one uses a regular lattice, as classically done for efficiency reasons. To recover on-grid propagation, this paper introduces a specific time discretization along each propagation directions and an original traversal algorithm to allow for scattering between different directions at common times. The algorithm involves only linear time interpolations so as to preserve the local nature of the Lattice Boltzmann Method. The direction quadrature follows the principles of the Discrete Ordinate Method. The relevance of the approach is illustrated on different two-dimensional problems and the results are compared to previously published numerical test-cases.