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On-the-Fly ROM-Based Acceleration of SI-DSA for Implicit Time Marching of the Radiative Transfer Equation

Ningxin Liu, Zhichao Peng

Abstract

In implicit time marching of the radiative transfer equation (RTE), the resulting linear systems are commonly solved using source iteration with diffusion synthetic acceleration (SI-DSA). Despite its widespread success, the performance of the DSA preconditioner may deteriorate when the RTE cannot be well approximated by its diffusion limit. Moreover, classical SI-DSA does not exploit low-rank structures of the solution manifold across time steps when the solution evolves smoothly. To address these limitations, we develop an on-the-fly reduced-order-model (ROM)-based acceleration for SI-DSA in implicit time marching of the RTE. Instead of relying on a diffusion approximation, the proposed approach constructs ROMs directly from the underlying kinetic formulation while exploiting low-rank structures in the temporal evolution of the solution. The method is fully offline-free and constructs ROMs to enhance both initial guesses and preconditioners on the fly during time marching. To handle streaming solution data, we design efficient and memory-lean ROM construction and adaptive update strategies based on dynamical mode decomposition, incremental low-rank singular value decomposition, and error indicators. Numerical experiments demonstrate that the proposed method consistently accelerates implicit time marching, delivering $1.4\times$ to $2.0\times$ speedup over classical SI-DSA while incurring only marginal overhead for ROM construction and updates.

On-the-Fly ROM-Based Acceleration of SI-DSA for Implicit Time Marching of the Radiative Transfer Equation

Abstract

In implicit time marching of the radiative transfer equation (RTE), the resulting linear systems are commonly solved using source iteration with diffusion synthetic acceleration (SI-DSA). Despite its widespread success, the performance of the DSA preconditioner may deteriorate when the RTE cannot be well approximated by its diffusion limit. Moreover, classical SI-DSA does not exploit low-rank structures of the solution manifold across time steps when the solution evolves smoothly. To address these limitations, we develop an on-the-fly reduced-order-model (ROM)-based acceleration for SI-DSA in implicit time marching of the RTE. Instead of relying on a diffusion approximation, the proposed approach constructs ROMs directly from the underlying kinetic formulation while exploiting low-rank structures in the temporal evolution of the solution. The method is fully offline-free and constructs ROMs to enhance both initial guesses and preconditioners on the fly during time marching. To handle streaming solution data, we design efficient and memory-lean ROM construction and adaptive update strategies based on dynamical mode decomposition, incremental low-rank singular value decomposition, and error indicators. Numerical experiments demonstrate that the proposed method consistently accelerates implicit time marching, delivering to speedup over classical SI-DSA while incurring only marginal overhead for ROM construction and updates.
Paper Structure (19 sections, 40 equations, 10 figures, 6 tables, 2 algorithms)

This paper contains 19 sections, 40 equations, 10 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Implicit discrete ordinates upwind discontinuous Galerkin method
  4. Matrix-vector formulation
  5. Source iteration with diffusion synthetic acceleration (SI-DSA)
  6. Motivation of our work
  7. On-the-fly reduced order model (ROM)-based acceleration for time-dependent RTE
  8. Dynamic mode decomposition with incremental SVD
  9. On-the-fly ROM-based acceleration of SI-DSA for implicit time marching
  10. Comparison with DMD-based preconditioner in mcclarren2022data
  11. Numerical results
  12. Two-material problem in 1D slab geometry
  13. Isotropic scattering
  14. Variable scattering problem
  15. Lattice problem
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Flowchart for On-the-fly ROM-based acceleration workflow. IG: initial guesses. PC: preconditioner. ROM-IG/PC: ROM-enhanced initial guesses/preconditioner.
  • Figure 2: Results for the two-material problem in Sec. \ref{['sec:1d-two-material']}. Left: $\rho$ at $T=1000$. Right: number of iterations required to convergence at each time step.
  • Figure 3: Density profile at t=2.5 with condition CFL=1 and $\sigma_s=1$. Left: solution by DMD enhanced SI-DSA, where we set $\epsilon_{\textrm{IG}}=10^{-9}$, $\epsilon_{\textrm{PC}}=10^{-6}$, and $\epsilon_{\textrm{UP}}=10^{-9}$. Middle: reference solution by SI-DSA. Right: the comparison of two solutions with $y=0$.
  • Figure 4: Number of iterations in each time step with condition $\sigma_s=1$. Left: CFL=0.25. Right: CFL=2.
  • Figure 5: Density profile at $t=2.5$ for the variable scattering problem in Sec. \ref{['sec:variable-scattering']}. Left: solution by DMD enhanced SI-DSA. Middle: reference solution by SI-DSA. Right: the comparison of two solutions with $y=0$.
  • ...and 5 more figures