The neutron transport equation in exact differential form

Abstract

Derived from the Boltzmann equation, the neutron transport equation describes the motions and interactions of neutrons with nuclei in nuclear devices such as nuclear reactors. The collision or fission effect are described as integral terms which arrive in an integro-differential neutron transport equation (IDNT). Only for mono-material or simple geometries conditions, elegant approximation can simplify the transport equation to provide analytic solutions. To solve this integro-differential equation becomes a practical engineering challenge. Recent development of deep-learning techniques provides a new approach to solve them but for some complicated conditions, it is also time consuming. To optimize solving the integro-differential equation particularly under the deep-learning method, we propose to convert the integral terms in the integro-differential neutron transport equation into their corresponding antiderivatives, providing a set of fixed solution constraint conditions for these antiderivatives, thus yielding an exact differential neutron transport equation (EDNT). The paper elucidates the physical meaning of the antiderivatives and analyzes the continuity and computational complexity of the new transport equation form. To illustrate the significant advantage of ENDT, numerical validations have been conducted using various numerical methods on typical benchmark problems. The numerical experiments demonstrate that the EDNT is compatible with various numerical methods, including the finite difference method (FDM), finite volume method (FVM), and PINN. Compared to the IDNT, the EDNT offers significant efficiency advantages, with reductions in computational time ranging from several times to several orders of magnitude. This EDNT approach may also be applicable for other integro-differential transport theories such as radiative energy transport and has potential application in astrophysics or other fields.