Efficient Hybrid Discontinuous Galerkin Methods for the Radiation Transport Equation with Isotropic Scattering and Diffusive Scaling

핵심 개념

The authors propose new asymptotic preserving (AP) hybrid discontinuous Galerkin (DG) methods for the radiation transport equation (RTE) with isotropic scattering and diffusive scaling. The methods employ heterogeneous polynomial spaces for different moments to maintain the AP property while significantly reducing the computational cost.

초록

The authors consider the steady-state and time-dependent radiation transport equation (RTE) with isotropic scattering and diffusive scaling. They focus on the spherical harmonic (PN) method for angular discretization and discontinuous Galerkin (DG) methods for spatial discretization.
The key insights from the asymptotic analysis are:

To retain the uniform convergence, it is only necessary to employ non-constant elements for the degree zero moment in the DG spatial discretization.
Based on this observation, the authors propose a heterogeneous DG method that uses polynomial spaces of different degrees for the degree zero and remaining moments, respectively. This maintains the AP property while reducing the number of unknowns.
To further improve the convergence order, the authors develop a spherical harmonics hybrid DG finite volume method, which preserves the AP property and convergence rate while significantly reducing the number of unknowns.

Numerical examples are provided to illustrate the effectiveness and accuracy of the proposed schemes.

통계

None.

인용구

None.

핵심 통찰 요약

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

by Cory D. Hauc... 게시일 arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10159.pdf

더 깊은 질문

How can the proposed hybrid DG/FV method be extended to handle more general boundary conditions beyond the periodic case considered in the paper

The proposed hybrid DG/FV method can be extended to handle more general boundary conditions by incorporating appropriate boundary treatments. For non-periodic boundary conditions, techniques such as the imposition of boundary conditions through penalty terms or the use of numerical fluxes can be employed. In the case of complex boundary conditions, such as mixed or Robin boundary conditions, specialized treatments like the introduction of boundary penalty terms or the implementation of boundary-fitted meshes may be necessary. Additionally, the use of high-order boundary schemes or the incorporation of boundary correction terms can enhance the accuracy and stability of the method for general boundary conditions.

What are the potential challenges in applying the heterogeneous DG and hybrid DG/FV methods to time-dependent problems or higher-dimensional settings

Applying the heterogeneous DG and hybrid DG/FV methods to time-dependent problems or higher-dimensional settings may pose several challenges. In time-dependent problems, the extension of the methods requires careful consideration of stability and accuracy issues related to time integration schemes. The treatment of time-dependent boundary conditions and the management of numerical dissipation in the presence of time-dependent solutions are crucial aspects to address. In higher-dimensional settings, the increase in computational complexity and memory requirements due to the higher dimensionality can pose challenges. Additionally, the construction of suitable mesh structures, the management of interpolation errors, and the optimization of computational resources become more intricate in higher-dimensional spaces.

Can the insights from the asymptotic analysis be leveraged to design efficient numerical methods for other types of kinetic equations beyond the RTE

The insights from the asymptotic analysis can indeed be leveraged to design efficient numerical methods for other types of kinetic equations beyond the Radiation Transport Equation (RTE). By understanding the diffusive scaling behavior and the asymptotic limits of the equations, similar hybrid DG/FV approaches can be developed for other kinetic models with diffusive scalings. The concept of employing heterogeneous polynomial spaces for different moments can be extended to other kinetic equations with diffusive regimes, allowing for the construction of efficient numerical schemes that preserve the diffusive limits while reducing computational costs. Furthermore, the analysis of convergence properties and the design of hybrid methods based on the asymptotic behavior can be applied to various kinetic models in physics, chemistry, and engineering where diffusive scaling plays a significant role.