Reformulation and evaluation of robust characteristic-based discretization for the discrete ordinates equation on structured hexahedron grids
Introduction
The discrete ordinates method (SN) is one of the classic deterministic numerical methods in discretizing the angular variable of the linearized Boltzmann transport equation. The spatial variable discretization is an essential step in numerical treatments for the SN equation, and countless studies center on spatial discretization methods under different types of grids. Reliable numerical simulations for the deep-penetration problems rely on proper mesh distributions and spatial discretization schemes of high accuracy and strong robustness.
It is generally thought that the spatial discretization schemes applied for nuclear installation shielding calculation can be classified into the types of finite difference, finite element, short characteristic and nodal methods. The behaviors of different schemes are problem-dependent and varied among the specific problems to be solved. Research shows that the convergence rate and the accuracy of spatial discretization methods depend on the smoothness of the exact flux solutions (Schunert and Azmy, 2015). The diamond difference method suffers from severe numerical diffusion (Mathews, 1999) and unphysical oscillations in the multi-dimensional geometry though it enjoys the second-order truncation error. Several nonlinear difference methods use the solution-dependent or angle-dependent fix-ups to abate the oscillations of global scalar flux distributions (Petrović and Haghighat, 1996). The finite element method (FEM) has been widely used for solving partial differential equations, and there is a large collection of derived numerical schemes in the particle transport community, such as linear discontinuous (Reed and Hill, 1973; Wareing et al., 2001; Adams, 2001), exponential discontinuous (Wareing and Alcouffe, 1996), corner balance (Adams, 1997) and so on. The excellent robustness in the thick diffusion limit makes certain FEM schemes available for radiative transfer problems. The short characteristic (SC) method (Lathrop, 1969) was developed based on the streaming properties of the transport equation. High-order SC and FEM schemes exhibit better properties in numerical diffusion than those of finite difference and nodal methods (Duo and Azmy, 2007). Furthermore, the potential cost of SC schemes' slope rotation fix-up for negative fluxes is less expensive than that of the FEM's matrix lumping technique. A unified FEM framework (Schunert et al., 2011) was presented to analyze the above-mentioned schemes under different trial functions and spaces. A comprehensive evaluation can be drawn that the SC discretization methods are of superior efficiency for the majority of conventional shielding problems with deep penetration and strong attenuation although FEM possesses outstanding accuracy and robustness for a broad range of transport problems.
The purpose of this work is to extend and to improve the SC schemes under a consistent architecture for the SN shielding calculation. And this work is a preliminary step for a SC-based p-adaptivity (Hall et al., 2017) algorithm under the octree discontinuous structured grids. Research on SC spatial discretization has a history of several decades. It is well known that the SC discretization splits the computational mesh into several sub-cells or sub-slices according to the different incident angular fluxes, and the flux spatial moments are constructed based on the equivalence with the characteristic solution of the transport equation. After Lathrop proposed a step or constant SC for 2D rectangle cells in 1969, simplified linear SC schemes (Larsen and Alcouffe, 1981; Childs and Rhoades, 1993) were developed in the XY and XYZ geometries, which adopted the approximated first-order spatial moments from the difference approximation solution and the linear nodal equations to reduce the computational cost. However, the modification of this kind ignores the slice-based physical property of the transport process, damages the moment conservation and may bring about the numerical diffusion. DeHart extended Lathrop's SC method to the arbitrary 2D polygonal meshes (DeHart et al., 1994), which was the first attempt to apply the SC discretization on the unstructured meshes. Based on the split-cell idea, a series of piecewise-constant-linear, linear and exponential schemes (Mathews and Minor, 1993; Mathews and Brennan, 1997; Miller et al., 1996) were developed by Mathews and his colleagues in the slab, XY and XYZ geometries. Walters and Wareing also developed a nonlinear characteristic method (Walters and Wareing, 1996) based on the information theory. At the same time, a 2D arbitrarily high-order transport method of the characteristic type (AHOT-C) method (Azmy, 1992) was presented by Azmy as a different category of the high-order characteristic methods. Some deficiencies and difficulties of the 3D AHOT-C scheme were overcome in the work of the AHOT-C-UG (Ferrer and Azmy, 2012) for unstructured tetrahedron grids.
On the basis of the existing SC discretization studies and the frame of the ARES transport code (Zhang et al., 2018), the constant (CSC), constant-linear (CLSC), linear (LSC), exponential (ESC) and slice-balance-approach-based (SBA) (Grove, 2005) difference-like schemes are developed based on Cartesian hexahedron grids. Numerical degradation and instability of SC discretization for void medium or materials with small cross sections are resolved using an exact volume moment integration. The slice-balance physical property is considered and handled about the particle transport process along a specific direction through the mesh. Numerical optimizations are proposed for the computational procedure to improve efficiency. The behaviors of these schemes are analyzed and compared with several finite difference methods. The remainder of this paper is organized as follows. The reformulations of the constant, linear and nonlinear SC schemes are described in section 2. The evaluations of the SC integral moments are given in section 3. Numerical verifications and comparisons are discussed in section 4 and some conclusions are drawn in section 5.
Section snippets
The short characteristic formulation
For simplicity, the time-independent monoenergetic transport equation after the SN angular discretization is considered for XYZ geometrywhere indicates the angle vector of the m-th discrete direction; indicates the angular flux of m-th direction at the position ; indicates the macroscopic total cross section; indicates the total source term including the scattering and fixed source. The discrete direction index m is omitted
Evaluation of the integral moments
For the SC schemes based on the polynomial expansion, Fn and Gn,m integral moments are proposed to package the algebra difficulties. The computational accuracy of the integral moments has a direct impact on the validity of these spatial discretization methods. Different algorithms are tested under a broad range of mesh optical thickness. For fast computing, several recursive relations are derived based on the integration by parts,
Numerical results
The development of characteristic-based schemes aims at improving computational efficiency and reducing unphysical oscillations for shielding problems as far as possible. The behaviors of these discretization methods are compared with the cell-average-based finite difference (FD) schemes on the benchmarks and the self-designed problems. First, their computational cost and time are evaluated and compared based on a one-group, fixed-source problem using the S16-order level symmetric quadrature
Conclusions
On the basis of the idea of short characteristic discretization and the angle-dependent spatial decomposition, the constant, linear, exponential short characteristic and the slice-balance-approach-based difference-like spatial schemes are reformulated and proposed for the discrete ordinates transport calculation on Cartesian hexahedron grids. Adopting the consistent basis functions and the same local coordinate system to solve the mesh angular flux, these characteristic-based discretization
Credit author statement
Cong Liu: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Bin Zhang: Writing - review & editing, Funding acquisition, Xinyu Wang: Data curation, Project administration, Liang Zhang: Software, Yixue Chen: Resources, Supervision, Funding acquisition.
Declaration of competing interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11975097), the Major National S&T Specific Program of Large Advanced Pressurized Water Reactor Nuclear Power Plant (2019ZX06005001) and the Fundamental Research Funds for the Central Universities (2019MS038).
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