The discrete ordinates method is well-suited to implementation with block-structured adaptive mesh refinement (AMR). AMR meshes group points into logically-rectangular patches, and provide the benefits of adaptivity without sacrificing the efficiency and geometric regularity of regular grids. In particular, these meshes preserve the directional ordering of points required for explicit ordinate transport sweeps. A number of algorithmic issues must be addressed to make such a method practical. These include sequencing of ordinates and grids for parallel execution, simultaneous solution of the transport equation on multiple levels of the grid hierarchy, implicit coupling to the fluid energy, and conservation of energy in a time-dependent context where different grid levels advance with different timesteps. The author discusses these and other issues and present example calculations in two and three spatial dimensions.

Advanced numerical simulations that use adaptive mesh refinement (AMR) methods have now become routine in engineering and science. Originally developed for computational fluid dynamics applications these methods have propagated to fields as diverse as astrophysics, climate modeling, combustion, biophysics and many others. The underlying physical models and equations used in these disciplines are rather different, yet algorithmic and implementation issues facing practitioners are often remarkably similar. Unfortunately, there has been little effort to review the advances and outstanding issues of adaptive mesh refinement methods across such a variety of fields. This book attempts to bridge this gap. The book presents a collection of papers by experts in the field of AMR who analyze past advances in the field and evaluate the current state of adaptive mesh refinement methods in scientific computing.

Contains papers presented at the October 1998 SIAM Workshop on Object Oriented Methods for Interoperable Scientific and Engineering Computing that covered a variety of topics and issues related to designing and implementing computational tools for science and engineering.

Adaptive Mesh Refinement (AMR) in conjunction with high order upwind finite difference methods has been used effectively on a variety of problems. In this paper we discuss an implementation of an AMR finite difference method that solves the equations of gas dynamics with two material species in three dimensions. An equation for the evolution of volume fractions augments the gas dynamics system. The material interface is preserved and tracked from the volume fractions using a piecewise linear reconstruction technique. 14 refs., 4 figs.

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