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Technical Papers

Fourier Convergence Analysis of the Infinite Homogenous Multigroup Time-Dependent Boltzmann Transport Equation Using Discrete Ordinates Formulation

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Pages 23-36 | Received 21 Jun 2016, Accepted 15 Sep 2016, Published online: 23 Feb 2017


Fourier analysis of the continuous infinite homogenous multigroup (MG) formulation is investigated in this paper for the time-dependent Boltzmann transport equation using discrete ordinates formulation. In addition, a continuous two-group (2G) and one-group (1G) Fourier analysis is performed to generate an analytical spectral radius and provide the basis for a theoretical analysis of the convergence. The discrete 1G formulation is then presented, and the theoretical analysis is found to predict the same spectral radius as the continuous 1G formulation. A typical pressurized water reactor pin cell problem with 47-group library is then homogenized with reflective boundary conditions, and the numerical spectral radius is calculated using the MPACT code. The theoretical predictions and the numerical results from the pin cell case agree very well and are found to have the following behavior: (1) The spectral radius is usually very close to unity for standard parameters for realistic transient application, (2) the spectral radius generally decreases as a function of inners per outer M, (3) the spectral radius generally decreases as a function of time-step size and then increases beyond unity for extremely small time steps, and (4) the spectral radius is almost constant as a function of the inserted reactivity. Good agreement is observed with the MG Fourier analysis. Finally, it is shown that the group sweeping coarse mesh finite difference method is theoretically and numerically very slow to converge the time-dependent neutron transport equation and that it is necessary to move the right-hand-side fission and transient source to the left-hand side and to solve the entire matrix form of the system.


This research was supported by the Consortium for Advanced Simulation of Light Water Reactors (, an Energy Innovation Hub ( for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy contract DE-AC05-00OR22725.

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