Transport Equation In Slab Geometry Research Articles

Positivity-preserving scheme with adaptive parameters for solving neutron transport equations in slab geometry

Positivity-preserving scheme with adaptive parameters for solving neutron transport equations in slab geometry

Density Transform Method for Particle Transport Problems in Spherical Geometry with Linearly Anisotropic Scattering

We develop the density transform method for treating particle transport problems in spherical geometry with linearly anisotropic scattering. We consider both, the interior and exterior problems of a homogeneous sphere and show that the transform satisfies an equation that resembles particle transport equation in slab geometry. The boundary conditions for these two problems are different. We also work out the density transform method for linearly anisotropic scattering for a region of arbitrary shape.

Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces

<p style='text-indent:20px;'>We establish some results regarding the existence of solutions to a nonlinear mono-energetic singular transport equation in slab geometry on <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-spaces with <inline-formula><tex-math id="M3">\begin{document}$ p\in (1,+\infty) $\end{document}</tex-math></inline-formula>. Both the cases where the boundary conditions are specular reflections and periodic are discussed.</p>

Error analysis and uncertainty quantification for the heterogeneous transport equation in slab geometry

Abstract We present an analysis of multilevel Monte Carlo (MLMC) techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients (cross-sections) are heterogenous random fields. To do this we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates that are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the one-dimensional space–one-dimensional angle slab-geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order $\mathcal{O}(h^{\eta })$ for some $\eta $ where $h$ is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. We also show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo (MC) and MLMC: for particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over MC. We provide numerical results supporting the theory.

Kershaw closures for linear transport equations in slab geometry II: High-order realizability-preserving discontinuous-Galerkin schemes

This paper provides a generalization of the realizability-preserving discontinuous-Galerkin scheme given in [3] to general full-moment models that can be closed analytically. It is applied to the class of Kershaw closures, which are able to provide a cheap closure of the moment problem. This results in an efficient algorithm for the underlying linear transport equation. The efficiency of high-order methods is demonstrated using numerical convergence tests and non-smooth benchmark problems.

Kershaw closures for linear transport equations in slab geometry I: Model derivation

This paper provides a new class of moment models for linear kinetic equations in slab geometry. These models can be evaluated cheaply while preserving the important realizability property, that is the fact that the underlying closure is non-negative. Several comparisons with the (expensive) state-of-the-art minimum-entropy models are made, showing the similarity in approximation quality of the two classes.

Calculation of eigenvalues for neutron transport equation using Henyey-Greenstein phase function in slab geometry

Eigenvalues are obtained for one-dimensional steady-state neutron transport equation in slab geometry using Henyey-Greenstein (HG) phase function. Firstly, HG phase function is inserted into neutron transport equation then eigenvalues are calculated for different values of collision parameters c and t parameters. All results are calculated for P9 and U9 approximation and these results compared each other.

Solving the criticality problem with the reflected boundary condition for the triplet anisotropic scattering with the modified FN method

Abstract One speed, time independent neutron transport equation in slab geometry can be solved with the triplet anisotropic scattering for the criticality problem with the reflected boundary condition. The value of the critical slab thickness is investigated numerically by using the Modified FN (HN) method. Case's eigenfunctions, normalization relation and orthogonality relations must be derived for this scattering in order to use the Modified FN method. Some selected values, which can be calculated from the criticality equation, can be tabulated. Also the term which comes from the Poincaré-Bertrand formula may be added to the orthogonality formula of the continuum eigenfunction.

Application of UN method to neutron transport equation in slab geometry using HG phase function

AbstractThe steady-state one speed neutron transport equation without sources is studied in slab geometry using Henyey-Greenstein scattering function. The critical thicknesses are calculated for various c (mean number of secondary neutrons per collision) and t scattering parameters using UN method and obtained values are compared with PN method. All the numerical results for different collision scattering functions and t parameters of UN and PN methods are presented in the tables.

Convergence of the modified discrete ordinates method for the anisotropic scattering transport equation

Criticality problem of nuclear tractors generally refers to an eigenvalue problem for the transport equations. In this paper, we deal with the eigenvalue of the anisotropic scattering transport equation in slab geometry. We propose a new discrete method which was called modified discrete ordinates method. It is constructed by redeveloping and improving discrete ordinates method in the space of L1(X). Different from traditional methods, norm convergence of operator approximation is proved theoretically. Furthermore, convergence of eigenvalue approximation and the corresponding error estimation are obtained by analytical tools.

High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling

In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection–diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro–macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit–explicit (IMEX) Runge–Kutta scheme in time equipped with a properly chosen implicit–explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of ε→0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.

Diffusion synthetic methods for computational modeling of one-speed slab-geometry transport problems with linearly anisotropic scattering

Two diffusion synthetic methods are described for computational modeling of one-speed, slab-geometry transport problems with linearly anisotropic scattering. These methods are referred to as diffusion synthetic methods, since the lower-order diffusion is used to simplify numerical solutions to the higher-order transport equation. In part I of this paper the word simplify is used in the sense of reducing the number of iterations to a prescribed stopping criterion; in other words, in the sense of accelerating the iteration on the scattering source in discrete ordinates (SN) calculations, by generating an improved initial guess. In part II, the word simplify is used in the sense of generating numerical results for the angular flux by solving analytically the first-order form of the transport equation in slab geometry with diffusion approximation for the scattering source integral terms. As with these two offered synthetic methods, special approximate boundary conditions are used in the diffusion equation to account for prescribed incident flux on the outer boundaries of the slab, including vacuum boundary conditions. Numerical results are given to illustrate the application of these two synthetic methods.

Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling

In this paper, some theoretical aspects will be addressed for the asymptotic preserving discontinuous Galerkin implicit-explicit (DG-IMEX) schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu, and T. Xiong, High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling, http://arxiv.org/abs/1306.0227, 2013, submitted] for kinetic transport equations under a diffusive scaling. We will focus on the methods that are based on discontinuous Galerkin (DG) spatial discretizations with the $P^k$ polynomial space and a first order implicit-explicit (IMEX) temporal discretization, and apply them to two linear models: the telegraph equation and the one-group transport equation in slab geometry. In particular, we will establish uniform numerical stability with respect to Knudsen number $\varepsilon$ using energy methods, as well as error estimates for any given $\varepsilon$. When $\varepsilon\rightarrow 0$, a rigorous asymptotic analysis of the schemes is also obtained. Though the methods and the analysis are presented for one dimension in space, they can be generalized to higher dimensions directly.

Improved Mixed and Hybrid Discretization of the Transport Equation in Slab Geometry

In this article we deal with a mixed and hybrid finite element method slab geometry discretization of the transport equation arising from the new variational formulation introduced in Cartier and Peybernes (2011). The aim of this study is to construct such a discretization by preserving the diffusion limit in the entire diffusive region, close to the boundaries, and for internal interface problems.

A polynomial analytical solution to the one-speed neutron transport equation in slab geometry under inherently verified Mark-Marshak boundary conditions

A polynomial analytical solution to the one-speed neutron transport equation in slab geometry under inherently verified Mark-Marshak boundary conditions

A model for calculation of forward isotropic scattering with application to transport equation in slab geometry

Abstract In order to develop a general procedure, the scattering kernel is described in a consistent manner. By using this scattering kernel, eigenvalues and eigenfunctions and critical slab thicknesses are computed with a conventional spherical harmonics method (PN). The basis of this method relates to the generating function of Legendre polynomials. This article has improved analytical methods and numerical values. Numerical results indicate that the critical thickness varies with forward isotropic scattering. Results are discussed and compared with isotropic and anisotropic results using various methods available in the literature. Eigenvalues and critical thicknesses are calculated using the new phase function instead of conventional expansions.

A Linear Algebraic Analysis of Diffusion Synthetic Acceleration for the Boltzmann Transport Equation II: The Simple Corner Balance Method

In this paper we apply the development and linear algebraic analysis of the diffusion synthetic acceleration method presented in [S. F. Ashby, P. N. Brown, M. R. Dorr, and A. C. Hindmarsh, SIAM J. Numer. Anal., 32 (1995), pp. 128–178] to a different spatial discretization. Our model equation is the monoenergetic, steady-state, linear Boltzmann transport equation in slab geometry. The discretization consists of a discrete ordinates collocation in angle and the simple corner balance method in space. By expressing diffusion synthetic acceleration in this formalism, asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various limiting cases, including the asymptotic diffusion limit. These results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux at the boundaries. Numerical results confirm the theoretical estimates.

Cellular neural networks (CNN) simulation for the T N approximation of the time dependent neutron transport equation in slab geometry

This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T N) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space–time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision). The complete algorithm could be implemented using very large-scale integrated circuit (VLSI) circuitry. The efficiency of the calculation method makes it useful for neutron transport calculations.

The slab albedo and criticality problem for the quadratic scattering kernel with the HN method

Abstract The one speed, time independent neutron transport equation for slab geometry with quadratic anisotropic scattering kernel is considered. Albedo, transmission factor and criticality thickness are calculated by the HN method. The numerical results obtained are listed for different selected parameters. It is shown that the method is concise and leads to fast converging numerical results.

Application of theUNapproximation to the neutron transport equation in slab geometry

AbstractThe critical slab problem of the one-speed and one dimensional neutron transport equation for an isotropic homogeneous medium was studied by using the Chebyshev polynomial approximation, i. e. UNmethod. The results obtained by the UNmethod for various c values, using the two type boundary conditions which are known as Mark and Marshak boundary conditions, are presented in tables. The tables also include the results obtained by the PNmethod for comparison.