# Stability of the diamond difference approximation in energy to the Spencer-Lewis equation of electron transport

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## Abstract

Consider the Spencer-Lewis equation (S-L) of electron transport in an azimuthally symmetric slab geometry setting with energy restricted to a finite interval. Further, S-L is subject to boundary conditions in the form of known incident particle fluxes at the slab faces. The one-dimensional diamond difference approximation is applied in the energy variable to the continuous slowing down term (i.e., the energy derivative) of S-L, This results in a semi-discrete system of integro-differential equations in the spatial and angular variables (D.E.S-L). The numerical stability of D,E.S-L as an approximation to S-L is demonstrated for solutions of D.E,S-L that belong to an L2 function space.

The system of integro-differential equations may be rewritten as a system of integral equations. Under certain reasonable conditions on the data, the existence-uniqueness of solutions of the integral equations in a Banach space of square integrable functions with weighted norms is established. This implies the existence of L2 solutions of the integral equations.

Under further assumptions on the data, the solutions of the integral equations are shown to be the required solutions of the integro-differential equations as well. Moreover, if the data are all bounded, the solutions are bounded.