A truly two-dimensional, asymptotic-preserving scheme for a discrete model of radiative transfer

For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly 2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [L. Gosse and G. Toscani, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 337-342] in the 1D case. Building on former results of Birkhoff and Abu-Shumays [J. Math. Anal. Appl., 28 (1969), pp. 211-221], it is possible to express 2D kinetic steady-states by means of harmonic polynomials, and this allows one to build a scattering S-matrix yielding a time-marching scheme. Such an S-matrix can be decomposed, as in [L. Gosse and N. Vauchelet, Numer. Math., 141 (2019), pp. 627-680], so as to deduce another scheme, well-suited for a diffusive approximation of the kinetic model, for which rigorous convergence can be proved. Challenging benchmarks are also displayed on coarse grids.