Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor correction

A class of Galerkin schemes for time-dependent radiative transfer

The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to preserve basic properties like exponential stability and decay to equilibrium also on the discrete level. We present the basic a-priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure like first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed PN-finite element time stepping scheme is demonstrated by computational results

Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations

The steady-state simplified Pn (SPn) approximations to the linear Boltzmann equation have been proven to be asymptotically higher-order corrections to the diffusion equation in certain physical systems. In this paper, we present an asymptotic analysis for the time-dependent simplified Pn equations up to n = 3. Additionally, SPn equations of arbitrary order are derived in an ad hoc way. The resulting SPn equations are hyperbolic and differ from those investigated in a previous work by some of the authors. In two space dimensions, numerical calculations for the Pn and SPn equations are performed. We simulate neutron distributions of a moving rod and present results for a benchmark problem, known as the checkerboard problem. The SPn equations are demonstrated to yield significantly more accurate results than diffusion approximations. In addition, for sufficiently low values of n, they are shown to be more efficient than Pn models of comparable cost.Comment: 32 pages, 7 figure

Optimal prediction for radiative transfer: A new perspective on moment closure

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.Comment: 15 pages; version 4: sections removed, major reformulation

Light propagation from fluorescent probes in biological tissues by coupled time-dependent parabolic simplified spherical harmonics equations

We introduce a system of coupled time-dependent parabolic simplified spherical harmonic equations to model the propagation of both excitation and fluorescence light in biological tissues. We resort to a finite element approach to obtain the time-dependent profile of the excitation and the fluorescence light fields in the medium. We present results for cases involving two geometries in three-dimensions: a homogeneous cylinder with an embedded fluorescent inclusion and a realistically-shaped rodent with an embedded inclusion alike an organ filled with a fluorescent probe. For the cylindrical geometry, we show the differences in the time-dependent fluorescence response for a point-like, a spherical, and a spherically Gaussian distributed fluorescent inclusion. From our results, we conclude that the model is able to describe the time-dependent excitation and fluorescent light transfer in small geometries with high absorption coefficients and in nondiffusive domains, as may be found in small animal diffuse optical tomography (DOT) and fluorescence DOT imaging

Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies

[EN] The diffusion approximation to the time-dependent Boltzmann transport equation gives accurate results for traditional nuclear reactor designs, but new reactor designs and new fuel elements require neutron transport methods. We develop a numerical approximation to the time-dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics P L equations, for odd L, and on the Backward Euler finite difference discretization of time. The resulting scheme can be written as a stationary form of diffusive second order PL equations. This allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem, using coarse spatial grids to further reduce memory requirements. This scheme is validated with several transient benchmarks, where the convergence properties are established and compared with the simplified PL approximation. A more realistic transient benchmark, based on the two-group C5 MOX problem, is finally introduced, showing the need of high order P L approximation for complex fuel geometries.This work was partially supported by the Spanish Agencia Estatal de Investigacion under project ENE2017-89029-P-AR, and the Generalitat Valenciana under project PROMETEO/2018/035. The authors express their gratitude to the anonymous reviewers for their suggestions and helpful comments.Capilla Romá, MT.; Talavera Usano, CF.; Ginestar Peiro, D.; Verdú Martín, GJ. (2020). Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies. Journal of Computational and Applied Mathematics. 375:1-21. https://doi.org/10.1016/j.cam.2020.112814S121375McClarren, R. G. (2010). Theoretical Aspects of the SimplifiedPnEquations. Transport Theory and Statistical Physics, 39(2-4), 73-109. doi:10.1080/00411450.2010.535088Capilla, M., Talavera, C. 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