Paul Nelson

Finite-Difference Approximations and Superconvergence for the Discrete-Ordinate Equations in Slab Geometry

A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors.

Calculation of Eigenfunctions in the Context of Integration-to-Blowup

The authors consider linear two-point boundary-value problems satisfying sign conditions of the type found in neutron transport problems. An algorithm is presented for approximating the eigenfunction associated with the smallest critical length of such a problem. The approximate eigenfunction is generated as the solution of a terminal-value problem for the linear differential system. The terminal data are given near the critical length and are computed from the value at the terminal point of the solution of the associated Riccati initial-value problem. Under appropriate conditions the approximation is shown to converge uniformly to the eigenfunction as the terminal value approaches the critical length. Practical considerations are discussed in the context of several numerical examples.

Theoretical Properties of One-Dimensional Discrete Ordinates

It is argued that there exist two distinct versions of the discrete-ordinates approximation to the linear transport equation. We term these Nystrom and interpolatory discrete-ordinates. The former has been studied extensively mathematically, while the latter seems to be used exclusively in practice. The two versions are compared in the simple case of monoenergetic transport in azimuthally symmetric one-dimensional slab geometry. It is shown that the Nystrom approximation scheme is stable and both consistent with and convergent to the transport equation, while many interpolatory schemes are inconsistent with the transport equation. From the viewpoint of numerical implementation, the Nystrom scheme points to the idea of choosing a different (presumably larger) set of quadrature points in the approximation to the streaming operator than is used in the scattering and fission operators. The previously calculated flux serves as an initial flux guess in the scattering term for one sweep through the space-angle m...

Convergence of a certain monotone iteration in the reflection matrix for a nonmultiplying half-space

Abstract A certain monotonically increasing nonlinear iterative procedure, first employed systematically by Shimizu and Aoki, has proven quite effective for computing the reflection matrix for a spatially homogeneous half-space, where the direction and energy variables are treated by suitable discrete approximations. In this paper it is shown that this procedure converges, provided only that the underlying half-space is nonmultiplying. (“Nonmultiplying” means that the maximum expected number of particles emerging from a collision does not exceed unity, where the maximum is taken over all energies and directions of the incident particles.) Furthermore, in this case it is shown that a certain norm of the approximate reflection matrices produced by the iterative process is bounded above by the maximum secondary scattering ratio.

Convergence of inner iterations for finite-difference approximations to the linear transport equation

Convergence of the inner iteration scheme to the solution of the linear transport equation is established using the concept of weakly submultiplying.

Invariant imbedding applied to homogeneous two-point boundary-value problems with a singularity of the first kind

It is shown that previous work of Elder can be used to extend the version of invariant imbedding due to M. R. Scott to homogeneous (vector) differential systems having a singularity of the first kind. The boundary conditions considered consist of existence (finite) at the singularity and specified values for some subset of the dependent variables at a second point. The important special case of a second-order equation is discussed in some detail. Computational considerations are discussed and numerical examples are presented.

An adaptive method for the numerical solution of Fredholm integral equations of the second kind. I. regular kernels

An adaptive method based on the trapezoidal rule for the numerical solution of Fredholm integral equations of the second kind is developed. The choice of mesh points is made automatically so as to equidistribute both the change in the discrete solution and its gradient. Some numerical experiments with this method are presented.

Stability of the upwind discrete energy approximation to the Spencer-Lewis equation of electron transport

Abstract Under suitable physically reasonable assumptions, stability and convergence results are obtained for the upwind finite-difference approximation in energy to the Spencer-Lewis equation of electron transport.

Existence and uniqueness for spherically symmetric linear transport

Abstract The purpose of this note is to establish an existence and uniqueness theorem for the spherically and azimuthally symmetric steady-state linear transport equation. The methods used are similar to those of Olhoeft1 for bounded three-dimensional geometry and of Nelson2 for linear transport in a slab. Our basic result contains the corollary that, for very general scattering laws, nonmultiplying transport of linear particles in a spherically and azimuthally symmetric situation necessarily is subcritical. This should be contrasted with the example due to Nelson2 of critical nonmultiplying linear transport in a slab. Our result also, when combined with that of Olhoeft1 (see also 3 Theorem 12 of Case and Zweifel3) establishes rigorously that the linear transport equation subject to spherically and azimuthally symmetric data has a solution that also enjoys these symmetries. Finally, we note that our basic technique should have application to the extension to spherical geometry of convergence results known...

Error analysis and convergence for the spectral synthesis method with interfaces

SummaryError estimates and convergence are studied for the Galerkin-type spectral synthesis method relative to the continuous-energy, continuous-space, time-independent neutron diffusion equation. The diffusion coefficient and total macroscopic cross section are both space and energy dependent, and interfaces may be present. The estimate is in an energy type norm. The basic result shows that the error is optimal in the sense of being of the same order as the error provided by the best approximation to the actual solution in the approximation space where the spectral synthesis solution is found.ZusammenfassungFehlerabschätzung und Konvergenz der Galerkin-ähnlichen Spektralsynthesenmethode, angewandt auf die zeitunabhängige Neutronendiffusionsgleichung mit kontinuierlicher Abhängigkeit von Ort und Energie, werden diskutiert. Der Diffusionskoeffizient und der makroskopische Gesamtquerschnitt hängen beide von Ort und Energie ab, wobei auch Zwischengrenzen möglich sind. Die Fehlerschätzung erfolgt in einer Energietypennorm. Das Grundergebnis zeigt, dass der Fehler optimal ist in dem Sinne, dass er von der gleichen Ordnung ist wie der Fehler entstanden durch die beste Näherung zur exakten Lösung im Approximationenraum, wo die Spektralsynthesenlösung gefunden wird.