Finite element methods are developed for the solution of the neutron diffusion equation in space, energy and time domains. Constructions of piecewise polynomial spaces in multiple variables are considered for the approximation of a general class of piecewise continuous functions such as neutron fluxes and concentrations of nuclear elements. The approximate solution in the piecewise polynomial space is determined by applying the Galerkin scheme to a weak form of the neutron diffusion equation. A piecewise polynomial method is also developed for the solution of first-order ordinary differential equations. The numerical methods are applied to neutron slowing-down problems, static neutron diffusion problems, point kinetics problems and time-dependent neutron diffusion problems. The uniqueness, stability and approximation error of the numerical methods are considered. The finite element methods yield high-order accuracy, depending on the degree of the polynomials used, and thereby permit coarse-mesh calculations. The conventional multigroup method, the Crank-Nicolson and the Pad6 schemes are shown to be special cases of the finite element methods. Numerical examples are presented which confirm the truncation error and demonstrate the utility of the finite element methods in reactor problems.
A class of finite difference methods called splitting techniques are presented for the solution of the multigroup diffusion theory reactor kinetics equations in two space dimensions. A subset of the above class is shown to be consistent with the differential equations and numerically stable. An exponential transformation of the semi-discrete equations is shown to reduce the truncation error of the above methods so that they beoome practical methods for two-dimensional problems. A variety of numerical experiments are presented which illusthate the truncation error, convergence rates, and stability of a particular member of the above class.
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