Applications of the pseudo-harmonics to the multidimensional kinetics

In this work we propose the application of the pseudo-harmonics in modal methods of spatial kinetics. The pseudo-harmonics, or the pseudo-modes, are the eigenfunctions associated to the leakage+removal operator in each group of energy of the steady-state diffusion equation. This operator is self-adjoint and it is not necessary to calculate the adjoints eigenfunctions. In the present work the pseudo-harmonics will be obtained starting from the equation of diffusion discretized by the coarse-mesh finite difference (CMFD). The modal methods approximate the time-dependent flux for an expansion in functions predetermined, that include the spatial dependence, with unknown expansion coefficients, that include the whole dependent part of the time. The obtaining of the system whose solution they are the coefficients of the modal expansion is done with aid of a function weight. Using the pseudomodes obtained in the steady state calculations, in the modal expansion as been the dependent functions of the space, we will show that the function weight can be chosen as being the own adjoint pseudo-modes. This calculation leads to a linear system of ordinary differential equations with the defined dimension for the number of pseudo-harmonics employed. The solution of the equations is built through the analytical integration of the equations of the precursor of delayed neutron. The proposed method was tested and it present good results, when compared with the direct method. (author)