Development and Verification of Neutron Transport Code by Applying GSP₃⁽⁰⁾ Theory

In the mathematical context of acquiring numerical solutions for the multi-dimensional neutron transport equation, the conventional simplified spherical harmonics (SP_N) methodology exhibits a fundamental theoretical limitation stemming from the fact that the requisite mathematical interface and boundary conditions are directly appropriated from the simplistic one-dimensional physical scenario. Consequently, the implementation of this specific mathematical treatment inherently possesses the potential to precipitate a circumstance wherein the overall computational precision of the spatial domain calculation is, in certain specific operational configurations, paradoxically rendered even inferior to the computational accuracy that would otherwise be achieved through the deployment of the standard neutron diffusion approximation theory. To systematically address and rectify this aforementioned fundamental theoretical deficiency, the international nuclear engineering and reactor physics research community has formally proposed and subsequently developed the comprehensive theoretical framework known as the generalized SP_N (GSP_N) theory. This GSP_N theoretical framework possesses the distinct and highly advantageous capability to flexibly provide a comprehensive hierarchy of multi-level mathematical approximations for detailed neutronic analysis. Among this diverse spectrum of available mathematical approximations, the specific mathematical formulation denoted as GSP₃⁽⁰⁾ rigorously represents the lowest-level hierarchical approximation applicable to the third-order spherical harmonics expansion of the fundamental neutron transport equation, while simultaneously maintaining the exact same system of partial differential equations as the conventional and widely utilized third-order SP_N (SP₃) mathematical model. Firmly predicated upon the rigorous mathematical foundations established by the aforementioned GSP₃⁽⁰⁾ theoretical framework, the present academic research endeavor has culminated in the successful development and systematic implementation of a novel, fully functional, two-dimensional neutron transport nodal computational code, formally designated as GSP₃⁽⁰⁾-NTS. This newly developed computational code has been meticulously architected to successfully achieve the complete implementation of a carefully unified numerical solution framework that simultaneously accommodates both the conventional SP₃ computational model and the advanced GSP₃⁽⁰⁾ mathematical model. Throughout the rigorous process of executing the numerical solution algorithm, this unified computational framework deliberately employs the generalized transverse integrated nodal (GTIN) methodological approach with the specific objective of appropriately and accurately treating the complex higher-order transverse leakage terms that are inevitably introduced into the mathematical system by the application of strictly rigorous interface and boundary conditions. Furthermore, in order to guarantee maximum algorithmic flexibility, the implemented computational program inherently possesses the sophisticated programmatic capability to dynamically alternate among three distinct spatial basis function profiles, specifically encompassing the flat basis function, the linear basis function, and the parabolic basis function, thereby ensuring that the computational methodology can comprehensively satisfy a highly diverse array of varied numerical simulation requirements depending on the specific application scenario. An exhaustive analysis of the detailed computational results generated by this executed code unequivocally demonstrates that the conventional SP₃ calculation methodology is fundamentally insensitive to the specific spatial distribution characteristics of the utilized basis functions. However, under computational scenarios characterized by the presence of strongly absorbing physical media, the k_eff unavoidably exhibits a highly significant and phenomenologically concerning negative deviation possessing a numerical magnitude of approximately −1 200 pcm. In stark computational contrast to this observed behavioral pattern, when the advanced GSP₃⁽⁰⁾ numerical solver is configured to adopt the linear spatial neutron current distribution profile, the resulting computational deviations corresponding to the calculated effective neutron multiplication factor under both weakly absorbing and strongly absorbing operational conditions are remarkably restricted to mere values of 197 pcm and −164 pcm, respectively, representing a highly desirable error reduction magnitude exceeding 85% when quantitatively compared against the baseline performance of the conventional SP₃ theoretical scheme. When systematically evaluated across a comprehensively diverse suite of standardized benchmark testing problems, both the statistically evaluated root-mean-square error and the recorded maximum absolute deviation associated with the detailed spatial neutron flux density distribution computationally determined by the GSP₃⁽⁰⁾ theoretical methodology are universally observed to be substantially lower than the analogous error metrics generated by the corresponding conventional SP₃ computational schemes. In addition to this overarching trend, it has been conclusively determined that the statistically evaluated root-mean-square error associated with the spatial neutron flux density distribution when specifically utilizing the parabolic spatial neutron current profile is demonstrably and significantly superior in terms of computational fidelity when directly compared to the numerical results yielded by the implementation of the strictly linear spatial neutron current profile. A rigorous and comprehensive analysis specifically dedicated to the evaluation of overall computational efficiency systematically reveals that, although the GSP₃⁽⁰⁾ mathematical model exhibits a certain measurable degradation in computational performance when operating on extreme problems, it introduces absolutely no additional computational burden when directed towards realistic reactor core problems, ultimately equating completely in computational efficiency with the classical SP₃ methodological approach. Furthermore, a detailed and meticulous spatial mesh sensitivity analysis provides incontrovertible evidence demonstrating that the GSP₃⁽⁰⁾ computational model successfully preserves a fully stable numerical convergence even when operating under an extremely coarse spatial computational grid, whereas, by direct comparison, the conventional SP₃ computational model unconditionally necessitates the deployment of a significantly finer spatial discretization mesh resolution in order to successfully guarantee the acquisition of a comparably stable numerical solution. Ultimately, the synthesized research findings systematically corroborate the definitive conclusion that the advanced GSP₃⁽⁰⁾ theoretical framework unequivocally demonstrates a profoundly superior computational performance when specifically tasked with the mathematical treatment of physical problems characterized by strongly absorbing media and highly heterogeneous geometric structures, whilst its inherent mathematical dependence on the specific dimensions of the spatial computational mesh is significantly and quantifiably lower than the corresponding mesh dependency exhibited by the conventional SP₃ framework. Although it is undeniably true that the rigorous implementation of the exact physical interface and physical boundary conditions within the GSP₃⁽⁰⁾ theoretical framework inevitably results in a marginally measurable escalation in the total time consumed by the iterative convergence process under extreme conditions, this specific computational disadvantage can be highly effectively mitigated, and the overall total computational resource overhead can be consequently profoundly reduced, by strategically capitalizing upon the tremendous inherent mathematical advantages fundamentally provided by its robust numerical adaptability to coarse spatial computational grids.