Convergence of the discrete ordinates method for the transport equation

Abstract

In the preceding, we have attempted to survey developments in the investigation of convergence properties and error bounds for the discrete ordinates approximations to the transport equation. This summary has been carried out in an abstract setting which, we believe, serves to unify and clarify much previous work in this area. The direction of future investigations is also clearly indicated.

While the first studies of convergence properties of the discrete ordinates methods were carried out nearly fifteen years ago, in an astrophysical setting, they have generally not come to the attention of workers interested in practical applications of the method to nuclear reactor computational problems. In fact, as recently as 1968, a standard work on reactor computing methods refers to the convergence of the discrete ordinates method as an unsolved problem.

A number of practical implications follow from the results surveyed above. For example, the equivalence of the spherical harmonics (P_N) and Gauss-quadrature methods for slab geometry transport problems (in the sense that the solutions agree at the quadrature points) establishes the convergence of, and provides error bounds for, the P_N solutions at the quadrature points.

Hopefully, future extensions of the results surveyed here to include three-dimensional systems and discretization of the spatial variable will provide convergence proofs and practical error estimates which will be of use even in the most complicated practical applications of discrete ordinates methods.

Supported by the U. S. Atomic Energy Commission at the Pacific Northwest Laboratory, Richland, Washington, and the Union Carbide Nuclear Division, Oak Ridge, Tenn.