Computational Methods For Strongly Coupled Radiative Transfer Problems

Radiative transfer problems, associated in applications of astrophysics physics, controlled thermonuclear fusion, usually are multi-scale and multi-material problems. Some regions of the physical models are optically thin low-Z medium and other regions are optically thick high-Z medium. Opacities can vary over the relevant frequency range by several orders of magnitude. In regions whose cells are optically thick and diffusive, efforts at designing discrete ordinates methods mainly meet difficulties for solving the coupled equations in a fast and accurate fashion.This thesis is focused on designing asymptotic preserving schemes for radiative transfer problems and fast iterative methods for radiative transfer equation. The main results in this thesis are the following:(1) Asymptotic analysis is applied to a discrete form of the time-dependent, energy-dependent, nonlinear radiative transfer problems in 1D cylindrical geometry, in which the spatial variable dis-cretized by the simple corner balance method, and the angle variable discretized by diamond difference method. We find that as mean free path goes to zero, the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation. The analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients, then the numerical method obtains an accurate solution. In 1D sphere geometry such analysis meets the same result. Note that in the previous works such analysis has only been applied to the planar-geometry problems.(2) A numerically asymptotic preserving simple corner balance method is constructed for solving radiative transfer equations on quadrilateral grids in R-Z geometry. The method proposed in Ref. [72] doesn't solve the strongly-coupled radiative transfer equation and material energy equation on the same grids, so numerical results indicate it's not an asymptotic preserving method, and show poor performance in regions whose cells are optically thick and diffusive. Modifying the method by solving the strongly-coupled radiative transfer equation and material energy equation both on the subcells, numerical results show its improved accuracy and robustness in thick diffusive regions.(3) Acceleration methods for multi-group radiative transfer problems are studied. A consistent linear multi-frequency-grey acceleration (LMFGA) scheme is developed for the Diamond-Differenced multi-group radiative transfer equations, and implemented in 1D radiation hydrody-namical code RDMG. Numerical results in practical problems show that the scheme presented is effective and robust, and is above 20 times faster than Source Iteration (SI). To avoiding the "consistent differencing" limitation for diffusion synthetic acceleration (DSA), the grey transport acceleration method (GTA) is generalized and recasted in terms of a preconditioned system that is solved with a Krylov method (GTAK). For problems on quadrilateral grids in R-Z geometry, the speedup of GTA method relative to SI is above 10, and GTA method is significantly more efficient than the GTAK method.(4) Acceleration methods for radiative diffusion calculations are compared and applied to practical problems. Synthetic acceleration scheme (LMFGA) for multi-group radiative diffusion equations is designed as the same way as that for multi-group radiative transfer equations, and implemented in 1D radiation hydrodynamical code RDMG. Numerical results in practical problems show that the scheme presented is effective and robust. Its speedup relative to SI is above 10 in 1D case. In addition, LMFGA is tested and recasted in terms of a preconditioned system that is solved with a Krylov method (LMFGK) in 2D case. Numerical results in R-Z geometry indicate the speedup of LMFGA method relative to SI is about 3, and LMFGA method is more efficient than LMFGK. When applied to 2D multi-material problems with strong material discontinuities, LMFGK method is even less efficient than SI.