A study of low order spherical harmonic closures for rapid transients in radiation transport

The time-dependent radiation transport has been solved using the spherical harmonics method. One problem with this method is that the scalar flux computed with the time-dependent spherical harmonics in 2D can become negative for rapid transients. In this dissertation, two schemes are suggested to overcome this problem. One relies on relaxing hyperbolicity of the spherical harmonics closure by using a quasi-static approximation and the other relies on removing linearity by making the P1 equation nonlinear.;A quasi-static approximation is normally made in P 1 theory to derive time-dependent diffusion theory, and this approximation maintains a positive flux. We have therefore explored such a quasi-static closure in P3 in 2D to see if a positive flux can result. The resulting method is relatively accurate, producing better results than much higher order PN methods, even while having far fewer unknowns. However the numerical results show that the scalar flux can still go negative in a pulsed source problem in 2D and this method is still not robust.;Next, using the eigenstructure of the variable Eddington factor equations to satisfy both the isotropic and beam-like limits we have derived a new Eddington factor as the simplest explicit polynomial of current-to-flux ratio. We have used the discontinuity structure and relaxation length as a way to compare the nonlinear Eddington factors. We have found that the existence of a minimum in the relaxation length implies the existence of discontinuity solution for nonlinear Eddingion closures. The numerical results show that our Nonlinear P1 closure produces very similar results with Maximum Entropy closure in 1D and remains positive for rapid transients in 2D. This method therefore provides a robust P1 method in 1 and 2D.