New computational technique based on Bernstein polynomials and higher order finite difference schemes for temporal and spatial reactor calculations

Abstract

In this research, two main objectives about solving the space time dependent diffusion equations are addressed. The first is to present higher order finite difference schemes at the core interior meshes and also at the points adjacent to the core boundaries, for spatial discretization. The second is to provide an accurate iteratively numerical method for temporal calculations based on convergent Bernstein polynomials. Bernstein series coefficients are estimated by using an easily computable recurrence relation. Estimation of the Bernstein recurrence relation requires nothing except the previous estimation of the series coefficients. Stability and convergence of the proposed method is discussed. It has proved that the method is conditionally stable and it has an exponential rate of convergence. Recently, we provided fourth and sixth order schemes to approximate the neutron leakage terms. These schemes provided highly accurate results but a large amount of these gained accuracies are lost due to using the classical second order finite difference formula at the core boundaries which uses three point stencils with order of accuracy OΔ2. Here, Lagrange’ polynomials are used to construct another second order scheme using five point stencils instead of three points at the core edges. It is proved that using central fourth order scheme at the interior meshes associated with the new scheme at the boundaries significantly improves the accuracy. Also, it is proved that using such finite difference schemes do not affect the system complexity and the system size remains unchanged. Homogenous and heterogeneous problems are considered to test the efficiency and the validity of the proposed method. In addition to the simplicity of the method algorithm, the results are so accurate that they can be compared with the recently methods that divided the same cores into thousands of fuel assemblies using mesh generator tools like GAMBIT or ICEM-CFD.