Transport synthetic acceleration methods for one-dimensional deterministic transport problems

Abstract

We construct and analyze three transport synthetic acceleration schemes. Our goal is the development of a rapidly convergent acceleration scheme that is robust and computationally efficient even for thick, diffusive problems. Additionally, we hope to develop a method that extends easily to multiple dimensions. The multi-level TSA scheme is a straightforward extension of the standard TSA scheme in which each acceleration step is itself accelerated with TSA. We find that our multi-level TSA scheme does not meet our requirements for corner-balance spatial discretizations. We show conclusively that a multi-level TSA scheme with an arbitrary number of levels cannot effectively attenuate highly oscillatory error modes for thick, diffusive problems. In response to the failure of the multi-level method, we have developed the [e] -scaled TSA method. In the [e] -scaled method, we are solving a low-order S₂ version of the high-order problem, with scaled cross sections. The cross sections and sources in the low-order problem are scaled in such a way that the scattering cross section in our acceleration equations vanishes. This particular scaling makes the low-order equations very easy to solve. For a corner-balance spatial discretization, we show with both a Fourier analysis and numerical results, that a simple scaling of the cross sections produces a divergent method for all problems except those with very thin cells. Further analysis of the failure of the [e] -scaled TSA method led to the development of a "consistent" [e] -scaled TSA method. We have determined that the failure of our original [e]-scaled method was due to problems at both cell interfaces and the problem boundary. The consistent method introduces interface terms that preserve relaxation lengths across cell interfaces and the problem boundary. The inclusion of these interface terms has led to an acceleration scheme that is very robust and efficient. We have verified the effectiveness of the consistent method for the simple-corner balance spatial discretization with numerical results.