Abstract
The one-dimensional nonlinear comer balance method is a new spatial discretization scheme for solving the transport equation on grids consisting of arbitrarily connected polygonal meshes. It is a conceptually simple method based on particle balance over half-cells. We describe the development and realization of this method in slab geometry with isotropic scattering. We prove that it is a positive method given positive sources. We show that the method is exact in purelyabsorbing and source-free half-space problems. We then compare the method to the linear comer balance and the nonlinear characteristics methods in a source-free deep-penetration problem. We then discuss the effects of downwardly-concave sources on the interior solution. We analyze the method in the thick, diffusive limit and show that the nonlinear comer balance method satisfies a discrete diffusion equation with Marshak boundary conditions. We further demonstrate that this discrete diffusion equation becomes the standard three-point finite difference approximation in the fine mesh limit. We demonstrate the lack of importance of the loss of superposition and Marshak boundary conditions by solving several simple transport problems. Finally, we show that the method displays fourth order truncation error through the use of a simple numerical experiment and offer some conclusions and suggestions for future work,
Castrianni, Christopher Lee (1994). A one-dimensional nonlinear corner balance method for solving the neutron transport equation. Master's thesis, Texas A&M University. Available electronically from
https : / /hdl .handle .net /1969 .1 /ETD -TAMU -1994 -THESIS -C355.