A Galerkin method for numerically solving the energy-dependent neutron transport equation

In the present investigation, Galerkin's method is applied to the first-order form of the steady-state energy-dependent neutron transport equation. The energy variable is treated separately from the direction and position variables yielding a set of multigroup equations which are a generalization of the standard multigroup equations. Numerical examples indicate greater accuracy is obtained using these generalized multigroup equations than using the standard multigroup equations. In addition, an error analysis is performed in which the energy variable is considered separately from the direction and position variables.

This method is applied to the one-dimensional plane, spherical, and cylindrical multigroup transport equations. Expansion functions and numerical integration rules are selected so that the inner pro ducts resulting in Galerkin's method are evaluated exactly and so that the flux density approximation is a continuous piecewise polynomial with respect to the position variable. In plane geometry, Galerkin's method using a piecewise linear approximation in the position variable leads to equations that are identical to the finite-difference discrete ordinates equations. In addition, using a piecewise parabolic approximation in Galerkin's method appears to give the same equations as a new fourth-order finite-difference method. It is demonstrated that these equations can be efficiently solved noniteratively rather than iteratively as is done presently. In spherical and cylindrical geometries, Galerkin's method does not yield equations that are equivalent to the finite-difference discrete ordinates equations for any order of approximation. However, it is likely that the equations obtained using Galerkin's method for these geometries can be solved noniteratively in an efficient manner similar to that for plane geometry.