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In recent years, the moving least square (MLS) approximation has been widely used for numerical analysis of scattered data. Many numerical methods for solving partial differential equations are developed based on the MLS approximation. The MLS method is ideal for problems where the geometry of the domain is complex. An example of this is the neutron diffusion calculation. The collocation method is known that it has fast convergence speed, however, it is unstable and non-robust in many cases. In particular, it is the reason resulting in the fluctuation of the solution at the position that is on overlapping boundary region. This drawback can be dealt with the meshless local Petrov-Galerkin weak form for nodes on the interface boundary. In this paper, a combination of the weak form of the meshless local Petrov-Galerkin (MLPG) and the collocation method are applied to solve the neutron diffusion equation. Trial functions employed in the weak form of MLPG are constructed via the MLS method. In most MLPG methods numerical integration is required. The exception to this is the case of the collocation method in which the test functions are Dirac delta functions. Therefore, if the shape function is not constructed precisely enough, within an acceptable error range, the second order derivative of the shape function will lack accuracy. One of the reasons for the inaccuracy of the shape function is singularity problems occurring in the process of constructing the shape function. In this study, a solution is introduced to eliminate the singularity problem, allowing us to obtain the derivative of the shape function with sufficient precision. Finally, neutron diffusion problems are implemented in a combination of the weak-form of the meshless local Petrov-Galerkin (MLPG) and the collocation method is used to evaluate the efficiency and accuracy.

The neutron flux spatial distribution in Boiling Water Reactors (BWRs) can be calculated by means of the Neutron Diffusion Equation (NDE), which is a space- and time-dependent differential equation. In steady state conditions, the time derivative terms are zero and this equation is rewritten as an eigenvalue problem. In addition, the spatial partial derivatives terms are transformed into algebraic terms by discretizing the geometry and using numerical methods. As regards the geometrical discretization, BWRs are complex systems containing different components of different geometries and materials, but they are usually modelled as parallelepiped nodes each one containing only one homogenized material to simplify the solution of the NDE. There are several techniques to correct the homogenization in the node, but the most commonly used in BWRs is that based on Assembly Discontinuity Factors (ADFs). As regards numerical methods, the Finite Volume Method (FVM) is feasible and suitable to be applied to the NDE. In this paper, a FVM based on a polynomial expansion method has been used to obtain the matrices of the eigenvalue problem, assuring the accomplishment of the ADFs for a BWR This eigenvalue problem has been solved by means of the SLEPc library. (C) 2016 Elsevier Ltd. All rights reserved., This work has been partially supported by the Spanish Ministerio de Eduacion Cultura y Deporte under the grant FPU13/01009, the Spanish Ministerio de Ciencia e Innovacion under projects ENE2014-59442-P, the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (FEDER) under project ENE2015-68353-P (MINECO/FEDER), the Generalitat Valenciana under projects PROMETEOII/2014/008, the Universitat Politecnica de Valencia under project UPPTE/2012/118, and the Spanish Ministerio de Economia y Competitividad under the project TIN2013-41049-P.

System thermal-hydraulic code RELAP5 is based on a two-fluid, non-equilibrium, and non-homogeneous hydrodynamic model for simulation of transient two-phase behavior. The code model includes six governing equations to incorporate the mass, energy, and momentum of the two fluids. In this paper, linear stability analysis is performed to check the ill-posedness of the RELAP5 specific two-fluid model (TFM) for all normal and accident conditions of a standard pressurized water reactor (PWR). The analysis gives information about the soundness of the model and identifies the range of parameters where the solutions obtained from the model will be numerically convergent. The linear two-phase fluid dynamic stability (by dispersion analysis) of the RELAP5 one-dimensional two-fluid model and numerical stability of the difference equation formulation (by Von Neumann method) is presented. The present analysis shows that the two-fluid model becomes ill-posed for some fluid conditions, where the results are less accurate, so sensitivity analysis plays an important role. The ill-posed nature implies that results thus obtained (by finite difference method) have to be interpreted carefully because of the sensitive nature of reactor safety analysis. It is also identified that the variation in various parameters (like slip ratio, system pressure, void fraction, and phasic velocities) can affect the error growth rates. It has been demonstrated that the basic system of one-dimensional two-phase flow equations, that possesses complex characteristics, exhibits unbounded instabilities in the short-wavelength limit and constitutes an improperly posed initial value problem. The semi-implicit numerical method, which is unconditionally stable for hyperbolic systems, becomes unstable for non-hyperbolic systems. For some of the fluid conditions, even after the introduction of artificial viscosity terms (in the difference equation formulation) that damp the high-frequency spatial components of the solution, are not sufficient for regularization of the two-fluid model. Thus, there is a need for the addition of newer terms, e.g. bubble collision, so that the existing incomplete model provides better results. It is also demonstrated that the basic TFM of RELAP5 with additional collision term makes the system unconditionally well-posed which are originally conditionally well-posed. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. It is found that the instability associated with the two-fluid model discretized system of equations is related to the instability associated with the ill-posedness of the two-fluid model but is quantitatively different.

Response sensitivity coefficients with respect to nuclide cross sections consist of two parts, explicit sensitivity coefficients and implicit sensitivity coefficients. The explicit sensitivity coefficients, which account the direct impact of cross sections on the responses through neutron transport equation, can be calculated efficiently with the classical Perturbation Theory. The implicit sensitivity coefficients, which account the indirect impact of cross sections on the responses through resonance self-shielding, are either omitted in most sensitivity analysis codes, or accounted for based on simple resonance-calculation methods which are not applicable for complex fuel designs. In order to expand the implicit sensitivity analysis method to wider application domain, a method based on the Generalized Perturbation Theory (GPT) is proposed in this paper to calculate the implicit sensitivity coefficients by using the subgroup method in the resonance self-shielding calculation. Based on the in-house-developed 2-D general-geometry method-of-characteristic neutron-transport code AutoMOC and subgroup resonance self-shielding code SUGAR, the proposed method has been implemented in the COLEUS code for the sensitivity and uncertainty analysis. Numerical analysis is then performed to investigate the impact of the implicit sensitivity coefficients of eigenvalue on non-resonance nuclide cross sections in two single-cell cases with different enrichments. The eigenvalue sensitivity coefficients predicted by the COLEUS code are consistent with those calculated by the direct-perturbation method, the reference solution. The results show that the implicit sensitivity has an important impact on both sensitivity and uncertainty in some analyzed cases.