Calculating Time Eigenvalues of the Neutron Transport Equation with Dynamic Mode Decomposition

Ryan G Mcclarren

doi:10.1080/00295639.2018.1565014

Abstract

A novel method to compute time eigenvalues of neutron transport problems is presented based on solutions to the time-dependent transport equation. Using these solutions, we use the dynamic mode decomposition to form an approximate transport operator. This approximate operator has eigenvalues that are mathematically related to the time eigenvalues of the neutron transport equation. This approach works for systems of any level of criticality and does not require the user to have estimates for the eigenvalues. Numerical results are presented for homogeneous and heterogeneous media. The numerical results indicate that the method finds the eigenvalues that contribute the most to the change in the solution over a given time range, and the eigenvalue with the largest real part is not necessarily important to the system evolution at short and intermediate times.

Highlights

In scientific computing we are used to taking a known operator and making approximations to it
dynamic mode decomposition (DMD) finds the eigenvalues that are important in the time dependent solution over the time scales considered and that are resolved by the time step size
From this figure we see that at different times the slowest decaying mode that the DMD estimates correspond with the modes that are important to the dynamics during a time interval

Summary

Introduction

In scientific computing we are used to taking a known operator and making approximations to it. It is possible to use the action of the operator to generate approximations rather than using the operator itself This is what is done in, for example, Krylov subspace methods for solving linear systems where the action of a matrix is used to create subspaces of increasing size that are used to find approximations to the solution. There have been improvements to deterministic α eigenvalue computation techniques that use specialized solvers to find positive and negative eigenvalues [6, 7, 8] or form the full discretization matrices to find eigenvalues [9] Most of these methods either find only the eigenvalue with the largest real part (the rightmost eigenvalue in the complex plane), or require an accurate estimate to find other eigenvalues.

Dynamic Mode Decomposition

Alpha Eigenvalues of the transport operator

Results for Plutonium Sphere

Heterogeneous Media

14.1 MeV Neutron pulse

Discussion