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Reduced order models for nonlinear radiative transfer based on moment equations and POD/DMD of Eddington tensor

Joseph M. Coale, Dmitriy Y. Anistratov

Abstract

A new group of reduced-order models (ROMs) for nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. The Eddington (quasidiffusion) tensor that provides exact closure for the system of moment equations is approximated via one of several data-based methods of model-order reduction. These methods are the (i) proper orthogonal decomposition, (ii) dynamic mode decomposition (DMD), (iii) an equilibrium-subtracted DMD variant. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be accurate even with very low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

Reduced order models for nonlinear radiative transfer based on moment equations and POD/DMD of Eddington tensor

Abstract

A new group of reduced-order models (ROMs) for nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. The Eddington (quasidiffusion) tensor that provides exact closure for the system of moment equations is approximated via one of several data-based methods of model-order reduction. These methods are the (i) proper orthogonal decomposition, (ii) dynamic mode decomposition (DMD), (iii) an equilibrium-subtracted DMD variant. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be accurate even with very low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

Paper Structure

This paper contains 17 sections, 52 equations, 19 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Thermal Radiative Transfer
  3. The Multilevel Quasidiffusion Method
  4. Method Formulation
  5. Discretization
  6. Model Reduction with Data-Informed Closures
  7. Projection of Data onto Low-Rank Subspaces
  8. Proper Orthogonal Decomposition
  9. Dynamic Mode Decomposition
  10. Equilibrium-Subtracted DMD
  11. Numerical Results
  12. Test Problem
  13. Data Analysis
  14. Performance of Low-Rank ROMs
  15. Breakout Time
  16. ...and 2 more sections

Figures (19)

  • Figure 1: Notations in the cell $i$.
  • Figure 2: Discrete grid functions shown on a sample 2D spatial mesh
  • Figure 2: Ranks $k$ for each approximate database corresponding to different values of $\xi_{\text{rel}}$ for the POD
  • Figure 3: F-C test solution for the material temperature $(T)$ and total radiation energy density $(E)$ over the spatial domain at times t=1, 2, 3 ns.
  • Figure 3: Ranks $k$ for each approximate database corresponding to different values of $\xi_{\text{rel}}$ for the DMD
  • ...and 14 more figures