A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

Journal Article · Mon Dec 24 00:00:00 EST 2018 · Nuclear Science and Engineering

DOI:https://doi.org/10.1080/00295639.2018.1525977· OSTI ID:1492610

Hammer, Hans Ruediger ^[1]; Morel, Jim Emanuel ^[2]; Wang, Yaqi ^[3]

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1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Texas A & M Univ., College Station, TX (United States)
2. Texas A & M Univ., College Station, TX (United States)
3. Idaho National Lab. (INL), Idaho Falls, ID (United States)

We present that second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage. Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal. In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. Finallyl, we show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: 89233218CNA000001; AC07-05ID14517
• OSTI ID: 1492610
• Report Number(s): LA-UR--18-23281
• Journal Information: Nuclear Science and Engineering, Journal Name: Nuclear Science and Engineering Journal Issue: 4 Vol. 193; ISSN 0029-5639
• Publisher: American Nuclear Society - Taylor & Francis
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Neutron transport
nonlinear diffusion acceleration
voids
weighted least squares