Transport Equation In Spherical Geometry Research Articles

Some Results Concerning the Singular Solution of the Conservative Transport Equation in Spherical Geometry

We examine in this work one of the exact solutions of the conservative transport equation for isotropic scattering in spherical geometry, specifically the solution that is singular at the origin and vanishes at infinity. Two representations are known for that solution: one expressed as an infinite divergent series that is derived from the spherical harmonics method and another given by an integral that results from the technique of integration along the particle path and is confirmed here by the method of characteristics. We establish a connection between these representations by showing that the Borel sum of the first reproduces the latter. We also examine computational aspects of the solution expressed in various forms and discuss some standing issues related to it.

A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell

A numerically stable version of the spherical harmonics (PN) method for solving the one-speed neutron transport equation with Lth order anisotropic scattering in a spherical shell is developed. Implementing a stable PN solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a PN solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the PN solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of N+1 Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard PN solution, as demonstrated by numerical results tabulated for some test cases.

Variance reduction method for particle transport equation in spherical geometry

This article is devoted to the design of importance sampling method for the Monte Carlo simulation of a linear transport equation. This model is of great importance in the simulation of inertial confinement fusion experiments. Our method is restricted to a spherically symmetric idealized design: an outer sphere emitting radiation towards an inner sphere, which in practice should be thought of as the hohlraum and the fusion capsule, respectively. We compute the importance function as the solution of the corresponding stationary adjoint problem. Doing so, we have an important reduction of the variance (by a factor 50 to 100), with a moderate increase of computational cost (by a factor 2 to 8).

Exact solution of the neutron transport equation in spherical geometry

Abstract Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using PN approximation which is widely used in neutron transport theory.

Criticality calculations with PN approximation for certain scattering parameters of Anlı-Güngör and Henyey-Greenstein phase functions in spherical geometry

Abstract PN approximation is applied to neutron transport equation in spherical geometry using Anlı-Güngör (AG) and Henyey-Greenstein (HG) phase functions. These phase functions are used for calculating critical radius and results of PN approximation obtained from AG and HG phase functions are compared with each other for different collision scattering function. The numerical results obtained from AG and HG phase functions are presented in the tables and these results are compared.

Least-Squares Finite Element Discretization of the Neutron Transport Equation in Spherical Geometry

The main focus of this paper is the numerical solution of the steady-state, mono-energetic Boltzmann transport equation for neutral particles through mixed material media in a spherically symmetric geometry. Standard solution strategies, like the discrete ordinates method, may lead to nonphysical approximate solutions. In particular, a point source at the center of the sphere yields undesirable ray effects. Posing the problem in spherical coordinates avoids ray effects and other nonphysical numerical artifacts in the simulation process, at the cost of coupling all angles in the PDE setting. In addition, traditional finite element or finite difference techniques for spherical coordinates often yield incorrect scalar flux at the center of the sphere, known as flux dip, and oscillations near steep gradients. In this paper, a least-squares finite element method with adaptive mesh refinement is used to approximate solutions to the nonscattering, one-dimensional neutron transport equation in spherically symmetric geometry. It is shown that the resulting numerical approximations avoid flux dip and oscillations. The least-squares discretization yields a symmetric, positive definite, linear system which shares many characteristics with systems obtained from Galerkin finite element discretization of totally anisotropic elliptic PDEs. In general, standard algebraic multigrid techniques fail to scale on non-grid-aligned anisotropies. In this paper, a new variation of smoothed aggregation is employed and shown to be essentially scalable. The effectiveness of the method is demonstrated on several mixed-media model problems.

Chebyshev polynomials expansion method for solving the one-dimensional transport equation in spherical geometry

Abstract Using certain well-known properties of Chebyshev polynomials, an efficient analytical approach to evaluate the Boltzmann equation is presented in one-dimensional spherical geometry. The procedure is based on the expansion of the angular flux in terms of a series of Chebyshev polynomials.

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

Since in many cases curvilinear geometry is more appropriate than cartesian geometry for precise modeling of the complex systems for reactor calculation, we have developed the spectral Green's function (SGF) method which is employed to obtain angular and scalar flux distributions in heterogeneous sphere geometry with isotropic scattering. In this study, we showed that the neutron transport problems of homogeneous spheres could be reduced to the solution of plane geometry equation. Finally, some results are discussed and compared with those already obtained by diamond difference scheme to test the accuracy of the results. The agreement is satisfactory. SGF method is very suitable for the numerical solution of the neutron transport equation with isotropic scattering.

P1 approximation of neutron transport equation in spherical geometry

P 1 approximation method is used to solve the spherical transport problems. This method is based on the analytic solution and the use of the P 1 approximation which is obtained by spherical harmonic method in the pseudo-slab equation. We also investigate diffusion coefficient in spherical geometry. Results are very accurate than classic diffusion theory.

P1 approximation of neutron transport equation in spherical geometry

P1 approximation method is used to solve the spherical transport problems. This method is based on the analytic solution and the use of the P1 approximation which is obtained by spherical harmonic method in the pseudo-slab equation. We also investigate diffusion coefficient in spherical geometry. Results are very accurate than classic diffusion theory.

The FN solution of the time-dependent neutron transport equation for a sphere with forward scattering

The one-speed time-dependent and stationary neutron transport equation in spherical geometry with forward scattering is considered. A formal equivalence between the transport equations for a critical and for a decaying system is established. By considering the pseudo-slab problem the scaled transport equation is solved using the F N method. Numerical values of radii for a critical and time-dependent systems are tabulated as a function of the scattering parameters and the fundamental decay constant. Some of the results are discussed and compared with others obtained using various methods. The results agree for four or five significant figures with the published results. It is shown that the F N method yields good numerical results for the problem considered. Finally, a few remarks about the effect of the forward anisotropy on the radius is also given.

Richardson Extrapolation Applied to Difference Methods for Numerically Solving the Neutron Transport Equation in Spherical Geometry

Two new difference schemes are derived for numerically solving the transport equation in spherical geometry. The first difference method is positive; i.e., the calculated fluxes are never negative....

A Petrov-Galerkin finite element method for solving the neutron transport equation

A finite element method using different trial and test spaces is introduced for solving the neutron transport equation in spherical geometry. It is shown that the widely used discrete ordinates method can also be thought of as such a finite element technique, in which integrals appearing in the difference equations are replaced by one-point Gauss quadrature formulas (midpoint rule). Comparison of accuracy between the new method and the discrete ordinates method is discussed, and numerical examples are given to illustrate the greater accuracy of the new technique.

A Semi-Analytic Solution to the Transport Equation in Spherical Geometry

A Semi-Analytic Solution to the Transport Equation in Spherical Geometry

A Semi-Analytic Solution to the Transport Equation in Spherical Geometry

A Semi-Analytic Solution to the Transport Equation in Spherical Geometry

Time-Dependent Moments of the Monoenergetic Transport Equation in Spherical and Cylindrical Geometry

The time-dependent spatial-Legendre moments for the monoenergetic transport equation in spherical geometry are expressed in terms of the half- and logarithmically weighted-Legendre moments of the flux in plane geometry. In cylindrical geometry, the moments are shown to be directly proportional to those in spherical geometry.

Directionally Discontinuous Harmonic Solutions of the Neutron Transport Equation in Spherical Geometry

An exact infinite-series solution to the one-speed neutron transport equation in spherical geometry is presented. The analysis is based on a partial-range variant of the spherical harmonics formali...

Parallel shooting solution of the neutron transport equation in spherical geometry

A direct and efficient parallel-shooting method is given for the solution of the spherical harmonics approximation to the neutron transport equation in spherical geometry with arbitrary anisotropic scattering and source. The special numerical difficulties due to the singularity at the center of the sphere are solved in a simple and efficient way. The algorithm “measures” the roundoff instability of the problem and eliminates it by performing a minimum of matrix transformations that ensure the linear independence of the matrix columns.

TwoGroup Neutron Transport Theory in Spherical Geometry

A set of normal modes for the two-group steady-state neutron transport equation in spherical geometry is constructed. The singular eigenfunction-expansion technique is then used to develop a rigorous solution to the isotropically emitting spherical shell-source problem in an infinite medium.

Green's Functions for the One-Speed Transport Equation in Spherical Geometry

Several problems in one-speed neutron transport theory for spherically symmetrical systems are discussed. The singular eigenfunction expansion technique is used to construct a solution for a specific finite-slab Green's function. This slab solution is then used to construct the finite-medium spherical Green's function by extending the point-to-plane transformation concept. For the general case, the expansion coefficients are shown to obey a Fredholm equation, and first-order solutions are obtained; however, in the infinite-medium limit the solution is represented in closed form. In addition, the solution for the angular density in an infinite-medium due to an isotropic point source is developed directly from the set of normal modes of the transport equation. A proof that the result so obtained obeys the proper source condition at the origin is given.