Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Neutrino transport in type II supernovae: Boltzmann solver vs. Monte Carlo method

We have coded a Boltzmann solver based on a finite difference scheme (S_N method) aiming at calculations of neutrino transport in type II supernovae. Close comparison between the Boltzmann solver and a Monte Carlo transport code has been made for realistic atmospheres of post bounce core models under the assumption of a static background. We have also investigated in detail the dependence of the results on the numbers of radial, angular, and energy grid points and the way to discretize the spatial advection term which is used in the Boltzmann solver. A general relativistic calculation has been done for one of the models. We find overall good agreement between the two methods. However, because of a relatively small number of angular grid points (which is inevitable due to limitations of the computation time) the Boltzmann solver tends to underestimate the flux factor and the Eddington factor outside the (mean) ``neutrinosphere'' where the angular distribution of the neutrinos becomes highly anisotropic. This fact suggests that one has to be cautious in applying the Boltzmann solver to a calculation of the neutrino heating in the hot-bubble region because it might tend to overestimate the local energy deposition rate. A comparison shows that this trend is opposite to the results obtained with a multi-group flux-limited diffusion approximation of neutrino transport. The accuracy of the Boltzmann solver can be considerably improved by using a variable angular mesh to increase the angular resolution in the semi-transparent regime.Comment: 19 pages, 17 figures, submitted to A&