Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Asymptotic Preserving, Hybrid Discontinuous Galerkin Methods the Radiation Transport Equation with Isotropic Scattering and Diffusive Scaling

Cory D. Hauck, Qiwei Sheng, Yulong Xing

TL;DR

This work develops asymptotic preserving discretizations for the radiative transport equation with isotropic scattering and diffusive scaling by combining spherical-harmonics (P_N) angular discretization with discontinuous Galerkin spatial discretization. A key insight is that AP can be achieved with heterogeneous polynomial spaces, using non-constant elements only for the degree-zero moment while higher moments use constant spaces, thereby dramatically reducing degrees of freedom without sacrificing the diffusion limit. To regain higher-order spatial accuracy in the diffusion regime, the authors introduce a hybrid DG/FV method that employs DG for the zeroth moment and second-order FV reconstructions for the remaining moments, preserving AP and improving convergence. Numerical experiments in 1D and 2D demonstrate robust AP behavior, accurate diffusion-limit approximations, and substantial memory savings compared with conventional DG approaches, making the methods attractive for multiscale transport problems.

Abstract

Discontinuous Galerkin (DG) methods are widely adopted to discretize the radiation transport equation (RTE) with diffusive scalings. One of the most important advantages of the DG methods for RTE is their asymptotic preserving (AP) property, in the sense that they preserve the diffusive limits of the equation in the discrete setting, without requiring excessive refinement of the discretization. However, compared to finite element methods or finite volume methods, the employment of DG methods also increases the number of unknowns, which requires more memory and computational time to solve the problems. In this paper, when the spherical harmonic method is applied for the angular discretization, we perform an asymptotic analysis which shows that to retain the uniform convergence, it is only necessary to employ non-constant elements for the degree zero moment only in the DG spatial discretization. Based on this observation, we propose a heterogeneous DG method that employs polynomial spaces of different degrees for the degree zero and remaining moments respectively. To improve the convergence order, we further develop a spherical harmonics hybrid DG finite volume method, which preserves the AP property and convergence rate while tremendously reducing the number of unknowns. Numerical examples are provided to illustrate the effectiveness and accuracy of the proposed scheme.

New Asymptotic Preserving, Hybrid Discontinuous Galerkin Methods the Radiation Transport Equation with Isotropic Scattering and Diffusive Scaling

TL;DR

This work develops asymptotic preserving discretizations for the radiative transport equation with isotropic scattering and diffusive scaling by combining spherical-harmonics (P_N) angular discretization with discontinuous Galerkin spatial discretization. A key insight is that AP can be achieved with heterogeneous polynomial spaces, using non-constant elements only for the degree-zero moment while higher moments use constant spaces, thereby dramatically reducing degrees of freedom without sacrificing the diffusion limit. To regain higher-order spatial accuracy in the diffusion regime, the authors introduce a hybrid DG/FV method that employs DG for the zeroth moment and second-order FV reconstructions for the remaining moments, preserving AP and improving convergence. Numerical experiments in 1D and 2D demonstrate robust AP behavior, accurate diffusion-limit approximations, and substantial memory savings compared with conventional DG approaches, making the methods attractive for multiscale transport problems.

Abstract

Discontinuous Galerkin (DG) methods are widely adopted to discretize the radiation transport equation (RTE) with diffusive scalings. One of the most important advantages of the DG methods for RTE is their asymptotic preserving (AP) property, in the sense that they preserve the diffusive limits of the equation in the discrete setting, without requiring excessive refinement of the discretization. However, compared to finite element methods or finite volume methods, the employment of DG methods also increases the number of unknowns, which requires more memory and computational time to solve the problems. In this paper, when the spherical harmonic method is applied for the angular discretization, we perform an asymptotic analysis which shows that to retain the uniform convergence, it is only necessary to employ non-constant elements for the degree zero moment only in the DG spatial discretization. Based on this observation, we propose a heterogeneous DG method that employs polynomial spaces of different degrees for the degree zero and remaining moments respectively. To improve the convergence order, we further develop a spherical harmonics hybrid DG finite volume method, which preserves the AP property and convergence rate while tremendously reducing the number of unknowns. Numerical examples are provided to illustrate the effectiveness and accuracy of the proposed scheme.
Paper Structure (21 sections, 4 theorems, 110 equations, 4 figures, 3 tables)

This paper contains 21 sections, 4 theorems, 110 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and problem setting
  3. Scaled radiative transfer/transport equations
  4. Equations in reduced geometry
  5. Time discretion and translation to steady-state form
  6. Spherical harmonic method for angular discretization
  7. Discontinuous Galerkin method for spatial discretization
  8. Asymptotic Analysis and Diffusion Limit
  9. Asymptotic analysis
  10. Diffusion limit
  11. A discontinuous Galerkin method with heterogeneous polynomial spaces for different moments
  12. A hybrid discontinuous Galerkin/finite volume method
  13. Numerical experiments
  14. One-dimensional P1 model
  15. One-dimensional P3 model
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume the assumptions as:rte_scale hold. For any fixed $\varepsilon>0$, the system of $P_N$ equations eq:rte_scale_Pn has a unique solution $\bm{u}\in\bm{W}$.

Figures (4)

  • Figure 1: $L^2$ errors of $\rho^h$ for the one dimensional ${P}_1$ model at $t = 0.05$.
  • Figure 2: $L^2$ errors of $\rho^h$ for the one dimensional ${P}_3$ model at $t = 0.05$.
  • Figure 3: $L^2$ errors of $\rho^h$ for the two dimensional $P_3$ model at $t = 0.05$.
  • Figure 4: Numerical solution $\rho^h$ for the two dimensional $P_3$ model at $t = 0.05$.

Theorems & Definitions (7)

  • Theorem 2.1
  • Lemma 2.2: Stability
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3