Spherical harmonics methods for thermal radiation transport.

Abstract

An implicit, spherical harmonic (<italic>P_n</italic>) method for solving thermal transport problems is developed. The method uses a high resolution Riemann solver to produce an upwinded discretization. The high resolution scheme introduces nonlinearities to the radiation transport operator to avoid the creation of artificial oscillations in the solution. By using a minmod limiter a quasi-linear approach to solving this nonlinear system of equations is developed. Through analysis and numerical results it is shown that the quasi-linear approach does suppress artificial oscillations and gives better than first order accuracy and is less computationally demanding than a fully nonlinear solve. The time integration methods considered are the backward Euler method and a high resolution time integration method. Also, reflecting boundary conditions for the <italic>P_n</italic> equations in three-dimensions are presented. It is shown that the standard Riemann solver is not robust in the diffusion limit. A fix is suggested that scales out the dissipation added by the Riemann solver as spatial cells become optically thick. The Green's function for the one-dimensional <italic>P</italic>₁ thermal transport equations with <italic>C_v</italic> &prop; <italic> T</italic><super>3</super> is derived. The Green's function is used to create the <italic>P</italic>₁ solution to a common benchmark and to a problem of an infinite, pulsed line source. The implicit method was able to produce robust results to thermal transport problems in one and two dimensions. The implicit approach allowed the numerical method to take times steps on the longer material energy time scale rather than the speed of light time scale. In two dimensional problems the <italic>P_n</italic> solutions contained negative radiation energy densities. These negatives caused the material temperature to become negative as well. The free-streaming limit of the <italic>P_n</italic> equations is explored and it is shown why in transient problems in multiple-dimensions the <italic>P_n</italic> solutions can have negative energy densities.