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Title: An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method

Abstract

High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still bemore » performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. As a result, the algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.« less

Authors:
 [1]; ORCiD logo [1]
  1. Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States)
Publication Date:
Research Org.:
Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Fusion Energy Sciences (FES)
OSTI Identifier:
1767817
Alternate Identifier(s):
OSTI ID: 1775687
Grant/Contract Number:  
SC0016515
Resource Type:
Accepted Manuscript
Journal Name:
Computer Physics Communications
Additional Journal Information:
Journal Volume: 264; Journal ID: ISSN 0010-4655
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Nodal discontinuous Galerkin method; Reconstruction; Convection diffusion equation; Computational efficiency; Unstructured; Triangle elements; High-energy-density hydrodynamics

Citation Formats

Song, Yang, and Srinivasan, Bhuvana. An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method. United States: N. p., 2021. Web. doi:10.1016/j.cpc.2021.107873.
Song, Yang, & Srinivasan, Bhuvana. An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method. United States. https://doi.org/10.1016/j.cpc.2021.107873
Song, Yang, and Srinivasan, Bhuvana. Sat . "An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method". United States. https://doi.org/10.1016/j.cpc.2021.107873. https://www.osti.gov/servlets/purl/1767817.
@article{osti_1767817,
title = {An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method},
author = {Song, Yang and Srinivasan, Bhuvana},
abstractNote = {High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. As a result, the algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.},
doi = {10.1016/j.cpc.2021.107873},
journal = {Computer Physics Communications},
number = ,
volume = 264,
place = {United States},
year = {2021},
month = {2}
}

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