A Transport Synthetic Acceleration method for transport iterations

Abstract

We present a family of Transport Synthetic Acceleration (TSA) methods to iteratively solve within-group scattering problems. A single iteration in these schemes consists of a transport sweep followed by a low-order calculation which is itself a simplified transport problem. We describe the development and the realization of the method for an isotropic source in XY geometry. We carry out a Fourier Analysis for a continuous set of equations and report TSA behavior. We show that a previously proposed TSA method is unstable in two dimensions but that our modifications make it stable and rapidly convergent. We follow the same procedure for descritized transport equations, using Step-Characteristics and two Bilinear Discontinuous methods, and find that discretization enhances TSA performance. We then propose to implement a Conjugate Gradient method on the low-order problem, to use a crude quadrature set in the low-order problem and to set the number of low-order iterations per transport sweep to a finite value. We prove that these features represent simple and efficient improvements to the method. We test TSA on a series of physical problems and propose a set of parameters for which the method behaves especially well. We further demonstrate that TSA achieves a substantial reduction in computational cost over Source Iteration, regardless of discretization parameters or materials and emphasize that this gain is an increasing function of the scattering ratio. We devote the final section to some conclusions and suggestions for future work.