A comparison of iterative methods for solving the time-dependent Boltzmann transport equation

F. Douglas Swesty

Dept. of Physics & Astronomy, SUNY at Stony Brook, Stony Brook, NY 11733


Abstract

The solution of the implicitly discretized time-dependent Boltzmann transport equation has traditionally relied on source iteration approach which derives from a splitting of the linear system. However, source iteration by itself offers poor convergence properties which has necessitated the development of a variety of techniques for accelerating the convergence of source iteration. The most popular of these acceleration techniques has been diffusion synthetic acceleration (DSA). However, the technique of source iteration + DSA still relies on the use of recursive "sweeps" to solve the linear systems. The recursive aspect of the problem makes it difficult to achieve scalability to large numbers of parallel processors. An alternative approach to solving the linear system is to attack the unsplit system with a standard Krylov subspace method such as GMRES or BiCGSTAB. A number of preconditioning strategies can be applied to speed convergence of these methods. The advantage of this approach is that the methods are readily parallelizable over large numbers of processors. We present the results of a comparative study of the numerical efficiency of these two methods for two popular discretization schemes: diamond-differencing and the simple corner balance method. In particular we study the behavior of these methods in one and two spatial dimensions with regard to both the size of the linear systems and the scalability of the parallel implementations.