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A discrete-ordinate weak Galerkin method for radiative transfer equation

Maneesh Kumar Singh

TL;DR

The paper addresses numerical approximation of the stationary radiative transfer equation (RTE). A discrete-ordinate weak Galerkin (DOWG) method is proposed, integrating discrete-ordinate angular discretization with a weak Galerkin spatial discretization. The authors prove a stability result and an a priori error bound of order ${h^{k+1/2}}$ (with angular-quadrature terms in 2D and 3D) and validate the theory through 2D numerical experiments using the Henyey–Greenstein phase function, showing the method is parameter-free and competitive with discrete-ordinate DG methods. The work demonstrates the viability of WG-based spatial discretization for RTE on general meshes and complex geometries, providing a solid theoretical foundation and practical performance benefits.

Abstract

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A priori error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out.

A discrete-ordinate weak Galerkin method for radiative transfer equation

TL;DR

The paper addresses numerical approximation of the stationary radiative transfer equation (RTE). A discrete-ordinate weak Galerkin (DOWG) method is proposed, integrating discrete-ordinate angular discretization with a weak Galerkin spatial discretization. The authors prove a stability result and an a priori error bound of order (with angular-quadrature terms in 2D and 3D) and validate the theory through 2D numerical experiments using the Henyey–Greenstein phase function, showing the method is parameter-free and competitive with discrete-ordinate DG methods. The work demonstrates the viability of WG-based spatial discretization for RTE on general meshes and complex geometries, providing a solid theoretical foundation and practical performance benefits.

Abstract

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A priori error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out.
Paper Structure (16 sections, 7 theorems, 81 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 16 sections, 7 theorems, 81 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The model problem
  3. Discrete-ordinate WG finite element method
  4. Discrete-ordinate method
  5. Semi-discrete problem
  6. Weak Galerkin finite element method
  7. Fully-discrete method
  8. Error analysis
  9. Stability result
  10. Error estimate
  11. Numerical discussion
  12. Numerical examples
  13. Comparison with DG methods
  14. DODG method for RTE model (\ref{['SRte1']})
  15. DODSD method for RTE model (\ref{['SRte1']})
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $u$ and $u^m$ are the solutions of the model problem (SRte1) and the semi-discrete problem (SRtde1), respectively. In the $2D$ case, where $h_{\theta}$ is sufficiently small. And, in the $3D$ case, we have where $C$ is positive constant depending on $r$ and the phase function $\Phi$.

Figures (1)

  • Figure 1: Source iteration

Theorems & Definitions (10)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Example 4.1
  • Example 4.2