Steady and transient solutions of neutron diffusion equations via computational methods based on Hartley series and higher order finite difference schemes

• Published in: Annals of nuclear energy 2024-06, Vol.201, p.110403, Article 110403
• Main Author: Hamada, Yasser Mohamed
• Format: Article
• Language: English
• Subjects: Hartley series; Higher order finite difference schemes; Neutron diffusion model; Taylor’s expansion; Vandermonde matrix
• ISSN: 0306-4549; 1873-2100
• DOI: 10.1016/j.anucene.2024.110403

•A novel numerical method based on Hartley series expansion is introduced.•New higher order finite difference scheme is introduced for the core boundary points.•Vandermonde system is solved to obtain the Hartley coefficients.•Convergence and stability of Hartley series method are studied. In this work, new accurate solutions based on Hartley series expansion for temporal calculations and higher order finite difference schemes for space derivative calculations are presented for the neutron diffusion systems. Throughout a novel use of a linear combination of Hartley series, a new temporal approximate solution is obtained. Determination of the series coefficients is a basic requirement to complete the reactor power calculations. Hartley series coefficients are determined by solving an algebraic system whose fundamental matrix is the famous Vandermonde matrix. Convergence and stability of Hartley series method is discussed. It is proved that the method is conditionally stable and a new criterion on the time step size is introduced. In previous work, we provided an accurate finite difference scheme with order of accuracy OΔ4 by employing five point stencils in each direction with equally distance Δ at the core interior mesh points. But a large amount of the gained accuracies is lost due to using second order finite difference scheme at the core boundaries. Here, a new formula for the points adjacent to the core boundaries is derived with the same order of accuracy OΔ4. This procedure significantly improved the accuracy of the steady state calculations. For homogenous reactor as an example, the new scheme reduces the errors to negligible amounts. The results are tested for the well-known benchmark transient problems such homogenous, TWIGL and LRA-BWR reactor cores for validation purposes. These cores are divided into just a few dozens or hundreds of fuel assemblies. Then, we compared our results with those obtained recently by the Improved SIDK, FLUENT, TASNAM and BRBFCM methods that divided the same reactor core into thousands of meshes by using mesh generation tools such GAMBIT and ICEM-CFD.