Least-squares derivation of extremum and weighted-residual methods for equations of reactor physics-1

A least-squares method of solving the first-order Boltzmann equation is given, and it is used to derive directly a maximum principle which is shown to be equivalent to the well-known maximum principles for the second-order even- and odd-parity Boltzmann equations. This surprising result is also derived explicitly from the Euler-Lagrange equation for the functional of the maximum principle, by using the properties of even and odd functions of directions to reduce this second-order equation to a first-order equation. The maximum principle for the first-order equation is illustrated geometrically using a suitable function space to obtain a weighted-residual method having weights differing from the classical Galerkin weights. The method employed to obtain a maximum principle for the first-order Boltzmann equation is illustrated further by the derivation of a maximum principle for a first-order initial-value problem. A finite element representation for an approximate solution is used to develop a first-order finite difference method which is compared with some classical difference methods. Finally complementary principles are given for the first-order initial-value problem, which can be used to provide upper and lower bounds for local characteristics of a finite difference solution. (author)