Keywords: time dependent neutron diffusion equation, block iterative methods, block preconditioners.

To improve the safety of nuclear power reactors it is necessary to develop fast and accurate plant simulators. Under general assumptions, the neutronic population inside a nuclear power reactor can be modelled by the time dependent neutron diffusion equation in the approximation of two energy groups [1]. To analyze the behaviour of Vodo-Vodyanoi Energetichesky Reactor (VVER) nuclear power reactors it is necessary to discretize this equation in a hexagonal mesh. The spatial discretization selected consists of a high order finite element method based on a triangular mesh which assumes that the neutronic flux can be expanded in terms of the modified Dubiner's polynomials [3,2]. Once this discretization has been selected, the semidiscrete version of the time dependent neutron diffusion equation is solved. Since the ordinary differential equations resulting of the discretization of diffusion equations are, in general, stiff, implicit methods are necessary. We have used a finite differences method, that needs to solve a large system of linear equations for each time step. Since the energy groups structure defines a natural partition of the matrix of the system into different blocks with good properties different methods to solve the linears systems are studied that use this block structure. In this way we have studied the performance of block iterative methods for the solutions of these systems of equations such as the block Jacobi and the block Gauss-Seidel methods combined with different variational acceleration techniques. Also we have proposed an inexpensive preconditioner for the system. The methods have been tested for the matrices obtained for a two-dimensional transient benchmark problem. For this case, we have observed that the most efficient method is the preconditioned Gauss-Seidel method. The performance of the method does not have a strong dependence on the variational acceleration technique used.

1

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

2

S. González-Pintor, D. Ginestar, G. Verdú, "High Order Finite Element Method for the Lambda Modes problem on hexagonal geometry", Annals of Nuclear Energy, 36, 1450-1462, 2009. doi:10.1016/j.anucene.2009.07.003

3

G.E. Karniadakis, S. Sherwin, "Spectral/hp Element Methods for Computational Fluid Dynamics", Oxford University Press, Oxford, 2005.

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