Singular Eigenfunction Expansion Research Articles

Numerical Caseology by Lagrange Interpolation for the 1D Neutron Transport Equation in a Slab

Here, we are concerned with a new, highly precise, numerical solution to the one-dimensional neutron transport equation based on Case’s analytical, singular eigenfunction expansion (SEE). While a considerable number of numerical solutions currently exist, understandably, because of its complexity even in one dimension, there are only a few truly analytical solutions to the neutron transport equation. In 1960, Case introduced a consistent theory of the SEE for a variety of idealized transport problems and forever changed the landscape of analytical transport theory. Several numerical methods, including the Fn method, were based on the theory. What is presented is yet another, called the Lagrange order N method (LNM) featuring the simplicity and precision of the Fn method, but for a more convenient and natural Lagrangian polynomial basis.

Singular Eigenfunction Expansion Solution for Time-Dependent Neutron Transport with Delayed Neutrons

We develop an analytic solution for time-dependent neutron transport with delayed neutrons using the singular eigenfunction expansion method. Our approach is based on a technique for solving time-dependent neutron-transport problems without delayed neutrons (Case and Zweifel, Linear Transport Theory, Addison-Wesley, 1967), which we effectively generalize to include the presence of delayed-neutron precursors. In particular, we obtain eigenfunctions composed of two parts: one corresponding to the neutron angular flux and one corresponding to the delayed-neutron precursor concentration. We further demonstrate that these eigenfunctions are complete. We also provide numerical results for an example problem.

The Infinite Medium Green’s Function of Monoenergetic Neutron Transport Theory via Fourier Transform

In 1960, Ken Case published his seminal work on the singular eigenfunction expansion for the Green’s function of the monoenergetic neutron transport equation with isotropic scattering. Previously, the solution had been found by Fourier transform as pole and branch cut contributions. It was apparent the two solutions were equivalent; however, showing equivalence for general anisotropic scattering was an unresolved challenge”until now. The motivation for revisiting the Green’s function solution is to resolve this issue by constructing a moments solution through analytical recurrence and application of Christoffel-Darboux formulas. While nothing more than Case’s singular eigenfunction expansion will result, the approach is new and follows Case’s original reasoning in applying separation of variables common to partial differential equations to solve the transport equation; that is, an equivalence to the singular eigenfunction expansion by Fourier transforms should indeed exist.

On Radiative Transfer in Dense Vegetation Canopies

This article presents a review of the special features exhibited by the transport of photons in dense vegetation canopies. The conservation of photons for particles moving in canopy media yields a linear Boltzmann Equation similar to conventional participating media. However, because of the turbid medium approximation where the atomic leaves are modeled as a gas cloud of infinitesimal plates with specified orientation, the overall system is not rotationally invariant. The latter affects the mathematical nature of the canopy equation and requires special numerical treatment to generate accurate numerical solutions. Here, the most common solution techniques are reviewed with special emphasis on two highly accurate methodologies, namely the Singular Eigenfunction Expansion, which is the basis of the F N method for canopy transport and the Analytical Discrete Ordinate (ADO) method. It is shown that the conventional ADO method can be easily extended to canopy transport problems by taking advantage of special symmetries exhibited by the intercept function and the area scattering function employed to describe the canopy optical properties.

Multigroup caseology in 1D via the Fourier transform

An alternative view of the Fourier transform inversion provides a new solution to the age-old problem of multigroup neutral particle transport equation in one-dimensional plane geometry. Through analytical continuation, the inversion contour shifts from the real line to accommodate pole contributions as discrete matrix eigenfunctions with continuum contributions appear from the branch cut. We recast the solution in terms of continuum singular matrix eigenfunctions to derive a singular eigenfunction expansion. Closure is an immediate consequence of the Fourier transform inversion. Finally, based on knowledge of the matrix weight factors and closure, we construct a singular eigenfunction expansion, incorporating orthogonality. The significance of this work is that a concise and consistent multigroup eigenfunction expansion emerges for anisotropic scattering that is as easily applied as in the one-group case.

A Neutron Transport Benchmark in One-Dimensional Cylindrical Geometry: Revisited

A new benchmark for monoenergetic neutron transport in one-dimensional cylindrical geometry is presented. In the past, several accurate benchmarks (i.e., numerical solutions) in cylindrical geometry, based on the singular eigenfunction expansion of the solution to the corresponding pseudoproblem, have appeared in the literature. In the new formulation, called the direct FN method in cylindrical geometry, we base the FN solution directly on the integro-differential equation satisfied by the pseudoproblem. Through appropriate projections, a straightforward FN formulation results in singular integral equations for both the flux and current. Enhanced by convergence acceleration, the FN approximation accurately reproduces published benchmark solutions for both fixed sources and criticality. Thus, we have developed an entirely pedagogical self-contained and highly accurate benchmark based on an alternative application of FN theory.

The effects of different expansions of the exit distribution on the extrapolation length for linearly anisotropic scattering

Abstract HN and singular eigenfunction methods are used to determine the neutron distribution everywhere in a source-free half space with zero incident flux for a linearly anisotropic scattering kernel. The singular eigenfunction expansion of the method of elementary solutions is used. The orthogonality relations of the discrete and continuous eigenfunctions for linearly anisotropic scattering provides the determination of the expansion coefficients. Different expansions of the exit distribution are used: the expansion in powers of μ, the expansion in terms of Legendre polynomials and the expansion in powers of 1/(1 + μ). The results are compared to each other. In the second part of our work, the transport equation and the infinite medium Green function are used. The numerical results of the extrapolation length obtained for the different expansions is discussed.

From Fourier Transforms to Singular Eigenfunctions for Multigroup Transport

A new Fourier transform approach to the solution of the multigroup transport equation with anisotropic scattering and isotropic source is presented. Through routine analytical continuation, the inversion contour is shifted from the real line to produce contributions from the poles and cuts in the complex plane. The integrand along the branch cut is then recast in terms of matrix continuum singular eigenfunctions, demonstrating equivalence of Fourier transform inversion and the singular eigenfunction expansion. The significance of this paper is that it represents the initial step in revealing the intimate connection between the Fourier transform and singular eigenfunction approaches as well as serves as a basis for a numerical algorithm.

Some comments on case eigenfunctions

Singular integral equations with Cauchy type kernels are considered on a real interval. It is assumed that near the end points of the interval the solution of the Riemann-Hilbert problem diverges at most logarithmically. By using such an “end-point condition” in a simple case of neutron transport in a infinite medium, the evaluation of the continuum coefficients of the singular eigenfunction expansion is carried out easily. Moreover, it results that such coefficients have a structure completely analogous to that of the discrete coefficients, showing the intimate relationship between Case's continuum and dis crete modes.

On computing radiative heat flux distributions using the [formula omitted] method

the exit intensities, heat fluxes internal to the medium must be the eigenfunction for the continuum eigenvalues and N is its comnuted from the nrimarv results associated with the norm. These functions are given by (cf. (4)) sin&lar eigenfun~tion-expansion. A computational problem is encountered in such evaluations. The purpose of this communication is to demonstrate how this problem may be resolved. The heat flux within an anisotropically scattering planar slab of finite optical thickness is given from the singular eigenfunction expansion technique by (IS)

On eigendistributions in linear transport theory

SynopsisAn attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.

Constructive solution of the linearized Boltzmann equation. Scalar BGK model equation

Abstract The following one-dimensional scalar equation arisingin the BGK model is studied. Methods of resolvent integration are used to obtain the so-called singular eigenfunction expansions for a certain Banach space of functions. Functional calculus for two related operators is then developed and application of this formalism is used to obtain solutions to the slip-flow problem.

On Multiregion Problems in Plane Geometry and the Use of Half-Range Orthogonality Relations

A general systematic method to reduce multiregion problems in plane geometry to regular integral equations for the coefficients of singular-eigenfunction expansions is proposed. The method is based on the half-range orthogonality relations of the eigenfunctions and is applicable in two-groups as well as one-group theory. The method is used to solve two problems in one-group theory for isotropic scattering: the Milne problem for a half-space bounded by a slab of dissimilar medium and a problem of neutron transmission through two slabs. Numerical results are reported for several sets of parameters.

Reduced Two-Dimensional Critical Problem of Finite Cylindrical Reactor

The critical problem is considered for finite cylindrical reactors. By applying the concept embodied in the asymptotic solution of the mono-energetic transport equation, it is possible to reduce the two-dimensional transport equation to an axially and radially reduced one-dimensional form. In essence, the idea is to approximate one of the two space dimensions by the assumption of the asymptotic distribution, and to extend the reactor medium and the asymptotic distribution toward infinity. The resulting reduced equation is solved by making use of the singular eigenfunction expansion method introduced by Case. It is found that the two-dimensional flux distribution can be obtained with high accuracy from the reduced equation, excluding exceptional cases of extremely poor conditions. Numerical results are presented for a wide range of reactor multiplication factor, and are compared with the results obtained by the variational method. It is found that there is only a small contribution of the boundary transient term brought to the axially and radially reduced problem. The error introduced in reducing the space dimension can be evaluated by the perturbation theory. Numerical results for various sizes of reactor show that the perturbation is not very large, and that it does not bring marked changes to the eigenvalues of the transport operators.

Extension of the Case formulas to L p. Application to half and full space problems

The singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Hölder-continuous functions to the function spaces Xp = {f‖μf (μ) - Lp}, where the expansions are now to be interpreted in the Xp norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with Xp sources and incident distributions.

Reduced Two-Dimensional Critical Problem of Finite Cylindrical Reactor

The critical problem is considered for finite cylindrical reactors. By applying the concept embodied in the asymptotic solution of the mono-energetic transport equation, it is possible to reduce the two-dimensional transport equation to an axially and radially reduced one-dimensional form. In essence, the idea is to approximate one of the two space dimensions by the assumption of the asymptotic distribution, and to extend the reactor medium and the asymptotic distribution toward infinity. The resulting reduced equation is solved by making use of the singular eigenfunction expansion method introduced by Case. It is found that the two-dimensional flux distribution can be obtained with high accuracy from the reduced equation, excluding exceptional cases of extremely poor conditions. Numerical results are presented for a wide range of reactor multiplication factor, and are compared with the results obtained by the variational method. It is found that there is only a small contribution of the boundary transien...

A new method for the calculation of the emergent distributions for the anisotropic slab albedo problem

A method is presented for the calculation of the emergent particle distributions of the albedo problem in one-speed transport theory with arbitrary anisotropic scattering. From a singular eigenfunction expansion of the solution of the albedo problem and the orthogonality properties of these eigenfunctions, a pair of coupled Fredholm integral equations of the second kind are obtained for the reflected and transmitted distributions. These integral equations are readily decoupled, and can be approximated by a set of linear equations from which very accurate numerical solutions are obtained.

Comments on “Application of singular eigenfunction expansions to the propagation of periodic disturbances in a radiating grey gas”

Comments on “Application of singular eigenfunction expansions to the propagation of periodic disturbances in a radiating grey gas”

Particle Transport in Spherical Media with a Central Black Cavity

A transform procedure is applied to the integral form of the Boltzmann transport equation to obtain the particle density in a spherically symmetric medium surrounding a central black absorber. The singular eigenfunction expansion technique is applied to provide a general solution, and boundary conditions are derived for the general equation for arbitrary multiplication and source distributions. Specific applications to the spherical Milne problem and a uniform source distribution are presented to demonstrate the application of the general transform technique.

Application of Singular Eigenfunction Expansions to the Propagation of Periodic Disturbances in a Radiating Grey Gas

The effect of thermal radiation on the propagation of small disturbances generated by the harmonic oscillation of a planar wall, either in position or temperature or both, is considered. By separation of variables, it is shown that the governing linearized equations can be reduced to a homogeneous integral equation that admits both regular and singular solutions to form a complete set; thus, the problem can be solved by the application of singular eigenfunction expansions analogous to those used in neutron-transport theory. An exact closed form solution is obtained for disturbed quantities in the flow and radiation fields. The exact solution shows that the disturbances consist of a damped continuum mode with infinite wave speed in addition to two discrete modes with finite wave speed. One of the discrete modes represents the “modified classical” wave whereas the other discrete mode plus the continuum mode corresponds to the radiation-induced wave. For sufficiently small Bouguer numbers, the discrete mode of the radiation-induced wave disappears. The newly found continuum mode damps faster than the discrete mode. Consequently, only the discrete modes persist away from the boundary. The differential approximation is found to predict the modified classical wave accurately but predicts the radiation-induced wave only approximately. The implications as well as the accuracy of the differential approximation are discussed and compared.