AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a...
Infinite media problems have been well studied in Flatland as well as spaces of general dimension @cite_20 @cite_4 @cite_15 @cite_13 @cite_14 @cite_18 @cite_26 @cite_5 @cite_6 and for beams @cite_23 . Some exact solutions have been presented for bounded @cite_25 and layered @cite_24 media, and the singular eigenfunctions for Flatland have been derived @cite_3 . However, to the best of the authors' knowledge, solutions to the classic Milne and albedo problems for the half-space and the @math -function have not been presented. @cite_19 , using an asymptotic analysis, have presented solutions to the classic Milne and albedo problems in two dimensions. The solutions make use of the Flatland equivalent of Chandrasekhar's @math -function, which is left as the solution to an integral equation. We present a complementary derivation of the Milne and albedo problem solutions using the Wiener-Hopf technique. In addition, we present new solutions and benchmark values for @math and provide some Monte Carlo comparisons for the albedo problem.
@cite_18 @cite_14 @cite_4 @cite_26 @cite_6 @cite_3 @cite_24 @cite_19 @cite_23 @cite_5 @cite_15 @cite_13 @cite_25 @cite_20
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AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a... Abstract Existing two-stream approximations to radiative transfer theory for particulate media are shown to be represented by identical forms of coupled differential equations if the intensity is replaced by integrals of the intensity over hemispheres. One set of solutions thus suffices for all methods and provides convenient analytical comparisons. The equations also suggest modifications of the standard techniques so as to duplicate exact solutions for thin atmospheres and thus permit accurate determinations of the effects of typical aerosol layers. Numerical results for the plane albedos of plane-parallel atmospheres (single-scattering albedo = 0.8, 1.0; optical thickness = 0.25, 1, 4. 16; Henyey-Greenstein phase function with asymmetry factor 0.75) are given for conventional and modified Eddington approximations, conventional and modified two-point quadrature schemes, the hemispheric-constant method and the delta-function method, all for comparison with accurate discrete-ordinate solutions. A new two-... Abstract Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method.