Application of Lie groups to discretizing nuclear engineering problems

A method utilizing groups of point transformations is applied to the three and four group time-independent neutron diffusion equations to obtain invariant difference equations for one-region and composite-region domains in one-dimensional Cartesian, cylindrical, and spherical geometries. Also, the theory behind this particular method will be discussed. A comparison of the invariant difference equations will be made to standard finite difference equations as well as to analytical results. From the analytical results, it will be shown that the invariant difference technique gives exact analytical solutions for the grid point values. The construction of invariant difference operators technique is also applied to the one-dimensional P 3 equations from neutron transport theory in Cartesian geometry, using the FLIP formulation, which allows PL equations to be written in the form of sets of coupled ordinary differential equations. The use of finite transforms will be examined to transform multi-dimensional problems into one-dimension where then the construction of invariant difference operators technique can be used to create difference equations. The solutions to the set of equations can then be transformed back into the multi-dimensional geometries. The use of finite transforms along with the construction of invariant difference operators technique is applied to a simple two-dimensional benchmark problem.; In addition, a method using groups of point transformations along with Noether's theorem is shown to generate a conservation law that can be used to create a two-term recurrence relation which calculates numerically exact Green's functions in one dimension for the time-independent neutron diffusion equation for Cartesian, cylindrical, and spherical geometries. This method will be expanded to constructing two-term recurrence relations for an arbitrary number of spatial regions, as well as detailing starting point values for type 2 and type 3 homogeneous endpoint boundary conditions. Finally, the method of constructing two-term recurrence relations will be applied to Green's function matrices for the one-dimensional time-independent neutron diffusion equation for Cartesian, cylindrical, and spherical geometries. In particular, two-term recurrence relations for the off-diagonal elements of the Green's function matrices will be derived, and the method is adapted to take into account discontinuities in the value of a function.