Krylov Subspace Method For2-dimensional Particle Transport Problem

Source iteration is a iterative process usually used to solve the particle transport problem. For physical system in which particles typically undergo few collisions, source iteration converges rapidly. However for physical system containing sub-regions that are optically thick and scattering-dominated, most of the particles undergo many collisions before being captured or leaking out, source iteration is inefficient and costly. Moreover, false convergence is a more difficult problem in association with this slow convergence.In this paper, we consider the popular Krylov subspace method. Firstly, we show how to formulate the total linear system in the Discrete-ordinates. Hence we explain that the source iteration is algebraically the first-order stationary iteration based on matrix-splitting techniques. Then we introduce the preconditioned Krylov subspace method and show how to choose preconditioners. To avoid too much large memory storage for coefficient matrix and preconditioners, we adopt a matrix-free technique. By this matrix-free technique, the memory (excluding a few more memory for vectors’s storage) needed in the preconditioned BiCGSTAB is as much as that of the source iteration. So the preconditioned BiCGSTAB methods can solve the same large scale problems as the source iteration. Numerical experiments show the number of iteration and CPU time for convergence of the preconditioned BiCGSTAB methods are much less than that of source iteration for the physical system containing subregions that are optically thick and scattering-dominated. In these difficult cases, the preconditioned BiCGSTAB methods can accelerate the convergence speed effectively, while for simple physical system where most of the particles undergo few collisions before being captured or leaking out, source iteration converges a little faster than the preconditioned BiCGSTAB methods.