Diamond Difference Research Articles

Application of the spectral Green's function method for numerically solving discrete ordinates problems in cylindrical geometry

Abstract The spectral Green's function (SGF) method is used to solve numerically the transport of neutrons in an infinite homogeneous cylindrical domain. In the method, the scattering of neutrons is assumed to be isotropic and the transport equation in cylindrical geometry is reduced to plane-like transport equation by using our reasonable approach. Then, the plane-like transport equation is solved by using the SGF method and the results are compared with those already obtained by the Diamond Difference (DD) scheme to check the accuracy of the results. The agreement of the results is good and it can be concluded that the SGF method is very effective for the numerical solution of the transport equation.

Error Comparison of Diamond Difference, Nodal, and Characteristic Methods for Solving Multidimensional Transport Problems with the Discrete Ordinates Approximation

Error norms for three variants of Larsen’s benchmark problem are evaluated using three numerical methods for solving the discrete ordinates approximation of the neutron transport equation in multidimensional Cartesian geometry. The three variants of Larsen’s test problem are concerned with the incoming flux boundary conditions: unit incoming flux on the left and bottom edges (Larsen’s configuration); unit incoming flux only on the left edge; unit incoming flux only on the bottom edge. The three methods considered are the diamond-difference (DD) method, the arbitrarily high order transport (AHOT) method of the nodal type (AHOT-N), and of the characteristic type (AHOT-C). The last two methods are employed in constant, linear, and quadratic orders of spatial approximation. The cell-wise error is computed as the difference between the cell-averaged flux computed by each method and the exact value, then the L1, L2, and L∞ error norms are calculated. The new result of this study is that while integral error norms, i.e., L1 and L2, converge to zero with mesh refinement, the cellwise L∞ norm does not. Via numerical experiments we relate this behavior to solution discontinuity across the singular characteristic. Little difference is observed between the error norm behavior of the methods in spite of the fact that AHOT-C is locally exact, suggesting that numerical diffusion across the singular characteristic is the major source of error on the global scale. Nevertheless, increasing the order of spatial approximation in AHOT methods yields higher accuracy in the integral error norms sense. In general, the characteristic methods possess a given accuracy in a larger fraction of the number of computational cells compared to nodal methods or DD.

Reduction of the Spatial Discretization Error in the Method of Characteristics using the Diamond-difference Scheme

In this paper, the diamond-difference (DD) scheme, which is commonly used in discrete-ordinate codes, is applied to the method of characteristics (MOC) to reduce the spatial discretization error of the flat flux approximation. Smaller spatial discretization error allows coarser background mesh division, which leads to smaller computational burden. Some theoretical considerations on the DD scheme are discussed to clarify the strength of this method. An absorption cross section weighted DD scheme (AWDD), which utilizes macroscopic absorption cross section to set the weight, is also discussed. The DD and AWDD schemes are implemented to AEGIS, which is a lattice physics code based on MOC. Then the AEGIS code is applied to two different benchmark problems whose spatial discretization errors are large. The calculation results indicate that from the viewpoint of spatial discretization error, the AWDD scheme is superior to the conventional MOC in which the step characteristics approximation is commonly used. Since incorporation of the AWDD scheme to current MOC codes is very simple, it will be a good candidate of spatial discretization method for MOC codes.

Discontinuous Galerkin finite element method applied to the 1-D spherical neutron transport equation

Discontinuous Galerkin finite element methods are used to estimate solutions to the non-scattering 1-D spherical neutron transport equation. Various trial and test spaces are compared in the context of a few sample problems whose exact solution is known. Certain trial spaces avoid unphysical behaviors that seem to plague other methods. Comparisons with diamond differencing and simple corner-balancing are presented to highlight these improvements.

Spectral Green's Function Method for Neutron Transport with Backward and Forward Scattering in 1‐D Slab and Spherical Geometry

In this work, a combination of forward and backward scattering is studied for the numerical solution of one‐dimensional neutron transport problems in a finite homogeneous slab and sphere. The scattering of the neutrons is assumed to be anisotropic for the slab and isotropic for the spherical geometry. Numerical solutions are carried out by using both the spectral Green's function (SGF) and the diamond‐difference (DD) methods to check the accuracy of the results. Calculated numerical results of cell‐edge scalar fluxes are presented in the tables.

Reduction of the Spatial Discretization Error in the Method of Characteristics using the Diamond-difference Scheme

In this paper, the diamond-difference (DD) scheme, which is commonly used in discrete-ordinate codes, is applied to the method of characteristics (MOC) to reduce the spatial discretization error of the flat flux approximation. Smaller spatial discretization error allows coarser background mesh division, which leads to smaller computational burden. Some theoretical considerations on the DD scheme are discussed to clarify the strength of this method. An absorption cross section weighted DD scheme (AWDD), which utilizes macroscopic absorption cross section to set the weight, is also discussed. The DD and AWDD schemes are implemented to AEGIS, which is a lattice physics code based on MOC. Then the AEGIS code is applied to two different benchmark problems whose spatial discretization errors are large. The calculation results indicate that from the viewpoint of spatial discretization error, the AWDD scheme is superior to the conventional MOC in which the step characteristics approximation is commonly used. Since incorporation of the AWDD scheme to current MOC codes is very simple, it will be a good candidate of spatial discretization method for MOC codes.

Analysis of Inherent Oscillations in MultidimensionalSNSolutions of the Neutron Transport Equation

Recent pressure vessel fast fluence calculations have revealed numerical difficulties (spatial oscillations) in the S{sub N} solutions, which have persisted in spite of mesh refinement. It is demonstrated that other shielding/deep-penetration applications may be affected; in fact, any S{sub N} solution in which the uncollided flux component is significant is likely to exhibit such difficulties. Test problems have been designed to characterize and understand numerical difficulties. Main analyses are performed using the diamond-difference (DD) scheme, which is linear and forms the basis for other (more complex) low-order differencing schemes. Other low-order differencing schemes (e.g., the DD with negative flux fixup and the {theta}-weighted) may partly remedy the situation by reducing the oscillations or by eliminating the oscillations at a cost of oversmoothing the results everywhere (e.g., the zero-weighted scheme). These schemes provide more robust solutions, but the inherent difficulties (although reduced) still remain. Types of discontinuities that trigger the oscillations are also examined; it is difficult to envisage an actual practical application free of such discontinuities. The magnitude of numerical difficulties (oscillations) and their practical relevance will depend on all S{sub N} model features, the differencing scheme being used, and the application requirements, but this study has shown that theymore » are inherent to multidimensional finite-difference S{sub N} algorithms.« less

Local analytical discrete ordinate method for the solution of the radiative transfer equation

This paper represents a new, accurate approach to solve radiative transfer equation for each discrete ordinates. The radiative transfer equations can be approximated as a linear system of partial differential equations if the source term is averaged within the domain of a finite control volume. Then these equations can be solved analytically by superposition for multi-dimensional problems. Conventional solution (diamond differencing) of the discrete ordinate method (DOM) may lead to negative fluxes, which is physically unacceptable. The proposed method (local analytical discrete ordinate method, LADOM) does not suffer from such a problem. The predictions of LADOM are well compared with that of zonal and conventional finite-difference methods. The proposed method is more accurate, free from oscillation and simpler to apply to multi-dimensional problems than the conventional finite-difference method.

Path Length Differencing and Energy Conservation of theSNBoltzmann/Spencer-Lewis Equation

It is shown that the SN Boltzmann/Spencer-Lewis equations conserve energy locally if and only if they satisfy particle balance and diamond differencing is used in path length. In contrast, the spat...

Spatial Convergence Properties of the Diamond Difference Method in x,y Geometry

It is shown, for a model numerical experiment, that the diamond difference (DD) solution of the x,y geometry discrete ordinates equations, with a fixed angular quadrature set, converges in the norm...

Numerical solution of the unsteady radiative transfer equation

In problems involving the transport of radiation through matter, it is necessary to solve the radiative transfer equation numerically. The scope for analytic solutions is restricted because of the complexity of the radiation-material interaction problem; this is particularly severe if the material is compressible. A study is made of various difference schemes within the framework of the discrete S N method (1) with a view to obtaining a well-behaved solution. In addition to the usual requirements of consistency, stability and accuracy, there are further desirable features. Firstly, it is preferable to keep in the difference scheme, the photon conservation property as expressed by the basic transport equation. Secondly, the solution should be non-negative, not only because such values are unphysical, but also because of their effect on the radiation-material coupling. Such a coupling is usually non-linear, and negative solutions can lead to instabilities in the numerical solution. Finally, it is preferable that the method should lead to simple calculations. Results are given for an example involving the radiative heating of a semi-infinite slab. It is shown that the diffusion limit imposes a particularly severe test, and the method used by Lathrop in the computer code DTF-IV (6) is favoured. A second example demonstrates a disturbing oscillatory behaviour arising from Diamond differencing which is basic to the DTF-IV method. A modification to this method designed to overcome this problem is examined.