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An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem

Lena Baumann, Lukas Einkemmer, Christian Klingenberg, Jonas Kusch

TL;DR

The paper addresses the high computational burden of solving the linearized Su-Olson thermal radiative transfer model by developing an energy-stable multiplicative dynamical low-rank discretization. It combines a multiplicative splitting $f = B g$, Legendre angular discretization, carefully stabilized spatial discretization, and an augmented BUG DLRA integrator to obtain energy stability and local mass conservation, with a proven hyperbolic CFL condition. The authors provide a rigorous energy-stability proof and mass-conservation results for the DLRA scheme, supported by numerical experiments that show accurate reproduction of full-order solutions with reduced rank and reduced computational cost. This work offers a robust reduced-order framework for radiative transfer problems and points toward extensions to nonlinear BGK-type models in future research.

Abstract

Computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide variety of kinetic partial differential equations is the concept of dynamical low-rank approximation (DLRA). In this paper, we consider the thermal radiative transfer equations with Su-Olson closure, leading to a linearized kinetic model. For the conducted theoretical and practical considerations we use a multiplicative splitting of the distribution function that poses additional challenges in finding an energy stable discretization and deriving a hyperbolic Courant-Friedrichs-Lewy (CFL) condition. We propose such an energy stable DLRA scheme that makes use of the augmented basis update & Galerkin integrator. This integrator allows for additional basis augmentations, enabling us to give a mathematically rigorous proof of energy stability and local mass conservation. Numerical examples confirm the derived properties and show the computational advantages of the DLRA scheme compared to a numerical solution of the full system of equations.

An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem

TL;DR

The paper addresses the high computational burden of solving the linearized Su-Olson thermal radiative transfer model by developing an energy-stable multiplicative dynamical low-rank discretization. It combines a multiplicative splitting , Legendre angular discretization, carefully stabilized spatial discretization, and an augmented BUG DLRA integrator to obtain energy stability and local mass conservation, with a proven hyperbolic CFL condition. The authors provide a rigorous energy-stability proof and mass-conservation results for the DLRA scheme, supported by numerical experiments that show accurate reproduction of full-order solutions with reduced rank and reduced computational cost. This work offers a robust reduced-order framework for radiative transfer problems and points toward extensions to nonlinear BGK-type models in future research.

Abstract

Computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide variety of kinetic partial differential equations is the concept of dynamical low-rank approximation (DLRA). In this paper, we consider the thermal radiative transfer equations with Su-Olson closure, leading to a linearized kinetic model. For the conducted theoretical and practical considerations we use a multiplicative splitting of the distribution function that poses additional challenges in finding an energy stable discretization and deriving a hyperbolic Courant-Friedrichs-Lewy (CFL) condition. We propose such an energy stable DLRA scheme that makes use of the augmented basis update & Galerkin integrator. This integrator allows for additional basis augmentations, enabling us to give a mathematically rigorous proof of energy stability and local mass conservation. Numerical examples confirm the derived properties and show the computational advantages of the DLRA scheme compared to a numerical solution of the full system of equations.

Paper Structure

This paper contains 18 sections, 6 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Thermal radiative transfer
  3. Discretization of the multiplicative system
  4. Angular discretization
  5. Spatial discretization
  6. Time discretization
  7. Energy stability
  8. Advection form
  9. Conservative form
  10. Dynamical low-rank approximation for the energy stable system
  11. Background on DLRA
  12. DLRA scheme for multiplicative Su-Olson
  13. Energy stability of the proposed low-rank scheme
  14. Mass conservation
  15. Numerical results
  16. ...and 3 more sections

Key Result

Lemma 1

Let $y,z \in \mathbb{R}^{N_x}$ with indices $i,j=1,...,N_x$. It holds Moreover, let $\mathbf D^{+}\in\mathbb{R}^{N_x \times N_x}$ be defined as Then, $\sum_{i,j =1}^{N_x} z_j D_{ji}^{xx} z_i = - \sum_{j=1}^{N_x} \left(\sum_{i=1}^{N_x} D_{ji}^+ z_i\right)^2$.

Figures (3)

  • Figure 1: Flowchart of the stable and conservative method \ref{['DLRA-scheme']}.
  • Figure 2: Top row: Numerical results for the solution $B(x)g(x,\mu)$ of the plane source problem at time $t_{\text{end}}=8$ computed with the full solver (left) and the multiplicative DLRA scheme (right). Middle row: Scalar flux $\Phi$ (left) and temperature $T$ (right) for both the full system and the multiplicative DLRA scheme. Bottom row: Evolution of the rank in time for the multiplicative DLRA method (left) and relative mass error for both methods (right).
  • Figure 3: Top row: Numerical results for the solution $B(x)g(x,\mu)$ of the Su-Olson problem at time $t_{\text{end}}=3.16$ computed with the full solver (left) and the multiplicative DLRA scheme (right). Middle row: Scalar flux $\Phi$ (left) and temperature $T$ (right) for both the full system and the multiplicative DLRA scheme. Bottom row: Evolution of the rank in time for the multiplicative DLRA method.

Theorems & Definitions (16)

  • Definition 1: Macroscopic quantities
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3: Total energy
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 6 more