Keywords: neutron diffusion equation, triangular mesh, collocation method, eigenvalue problem, lambda modes, Fekete points, Dubiner polynomials, VVER reactors.

To improve the design and the safety of nuclear power reactors it is necessary to develop fast and accurate plant simulators. One of the models of the nuclear plant simulators is the Lambda modes equation, [1]. We will consider this equation in the two energy groups approximation. This is a differential eigenvalue problem. The dominant eigenvalues of this problem and their corresponding eigenfunctions have been used successfully to develop modal methods to integrate the time dependent neutron diffusion equation [2]. Thus, it is interesting to develop fast and accurate methods to obtain these modes.

Most of nuclear power plants under construction are Russian designed reactors VVER. These reactors are pressure water reactors (PWR) different from European or American PWR reactors in the geometry of their fuel elements: in VVER reactors the fuel elements are hexagonal prisms, while in PWR reactor they are rectangular prisms.

To discretize the problem for reactors of the VVER type, we divide each hexagon of the mesh into six equilateral triangles and use a method to discretize partial differential equations adapted to triangular cells. For nuclear reactors the spatial mesh is naturally defined by the structure of the core, it is interesting to use a method that uses a fixed mesh and increases its precision without changing this mesh. Nodal collocation methods are methods based on approximating the solution of the problem as a truncated expansion in terms of certain basis of the polynomials. The accuracy obtained with this kind of method is controlled by the number of polynomials considered in the expansion.

Different spectral methods are developed for the neutron diffusion equation using triangular meshes. These methods are based on the expansion of the neutronic flux for each node in terms of a basis of polynomials. The relation between the different nodes is established by means of the boundary conditions and continuity conditions on a set of collocation points, the Fekete points, [4].

We have chosen two different basis of polynomials, the standard polynomials, and the Dubiner's polynomials [3]. We have considered a point-wise version of the neutronic diffusion equation, obtaining a set of balance equations evaluating, for each node of the mesh, the neutron diffusion equation for a set of interior Fekete points of the right triangle. The relation among the different triangles of the mesh is imposed as a set of constrains given by the boundary and continuity conditions at the faces of the different triangles. An integral version of the spectral method has been also considered, obtaining the balance equations from a moments-like version of the neutron diffusion equation on each of the triangles of the mesh. The constrains considered have been the same as the ones used for the point-wise methods.

To test the performance of the spectral methods, we have calculated the dominant Lambda modes for two different bidimensional reactors. First, we have considered a homogeneous rectangular reactor discretized using a triangular mesh, since this problem admits an analytical solution. The second case analysed has been the bidimensional benchmark problem known as IAEA reactor without reflector. This last problem is a typical problem to test neutronic codes based on hexagonal geometry.

From these calculations, we conclude that the spectral methods succeed in calculating a set of dominant Lambda modes for the reactor. This feature is new for neutronic codes, since the existing codes only calculate the fundamental mode.

We have to remark that, for large problems as the IAEA reactor core the integral methods present a quite faster convergence with the maximum degree of the polynomial expansion K, than the one presented by the point-wise methods.

1

W.M. Stacey, "Nuclear Reactor Physics", John Wiley & Sons Inc., New York, 2001.

2

R. Miró, D. Ginestar, G. Verdú, D. Hennig, "A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis", Annals of Nuclear Energy, 29, 1171-1194, 2002. doi:10.1016/S0306-4549(01)00103-7

3

M. Dubiner, "Spectral methods on triangles and other domains", J. Sci. Comput., 6, 345-390, 1991. doi:10.1007/BF01060030

4

M.A. Taylor, B.A. Wingate, R.E. Vincent,"An algorithm for computing Fekete points in the triangle", SIAM J. Numer. Anal., 38, 1707-1720, 2000. doi:10.1137/S0036142998337247

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