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The problem of spatial approximation becomes very important in the solution of neutronics problems with coarse spatial grids, in particular, in the calculations of fuel assemblies of fast reactors (for instance, BN-800 and BN-1200 reactors) with computational cell in the form of hexagonal prism. “Weighted diamond difference” (WDD) schemes are the most widely used among the finite difference schemes for the neutron and gamma-ray transport equation solution. They are efficient from the viewpoint of ease of their implementation and associated CPU time expenditures. However, some drawbacks of these schemes are manifested when they are applied to solve the above described problems. Diamond difference scheme (DD) having second-order approximation (the best for this class of schemes) does not possess the properties of positivity and monotonicity. This is the reason why negative values and non-physical oscillations are often present in the solutions obtained. “Step” scheme (St), which is free from the disadvantages of the diamond difference scheme, has accuracy of only the first order. In connection with the need in high-accuracy calculations its use appears to be inefficient. There exist algorithms for correction of negative values, as well as adaptive (AWDD) schemes aimed both at the reduction of the level of oscillations and at the obtaining positive solutions. However, these algorithms negatively affect the order of approximation, and schemes of the first – second order of accuracy are discussed in such cases. Besides that, for adaptive schemes there exists the problem of correct selection of parameters of the scheme. The evident way to escape such situation with simultaneously enhancing quality and accuracy of the calculation is to select a fine mesh. In case of calculation of fuel assemblies of fast reactors spatial grid represents an arrangement of rectangular prisms with regular hexagons forming their bases (in such cases reference is made to HEX-Z-geometry). Therefore, hexagonal cells can be divided into rhomb-shaped cells (three rhombs per one hexagon; 12 rhombs per one hexagon, etc.). Diamond scheme is applied for the grids consisting of rhombs thus obtained. Because of the smaller cell size as compared with original cell size, the drawbacks inherent to this scheme will not be pronounced. Triangular grid can also be used. A different approach for the solution of the above indicated problem is to develop computational methods with enhanced order of accuracy without increasing the number of computational points. Nodal method is one of such methods. Expansion of unknown function inside the node (elementary volume with constant properties) in basis functions with subsequent calculation of expansion moments constitutes the basis of any nodal method. Nodal SN-method in HEX-Z geometry will be discussed in the present paper. |
