Volume 13, Issue 4
Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

Haiyang Gao, Z. J. Wang & H. T. Huynh

Commun. Comput. Phys., 13 (2013), pp. 1013-1044.

Published online: 2013-08

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  • Abstract

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

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@Article{CiCP-13-1013, author = {}, title = {Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations}, journal = {Communications in Computational Physics}, year = {2013}, volume = {13}, number = {4}, pages = {1013--1044}, abstract = {

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.020611.090312a}, url = {http://global-sci.org/intro/article_detail/cicp/7262.html} }
TY - JOUR T1 - Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations JO - Communications in Computational Physics VL - 4 SP - 1013 EP - 1044 PY - 2013 DA - 2013/08 SN - 13 DO - http://dor.org/10.4208/cicp.020611.090312a UR - https://global-sci.org/intro/article_detail/cicp/7262.html KW - AB -

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

Haiyang Gao, Z. J. Wang & H. T. Huynh. (2020). Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations. Communications in Computational Physics. 13 (4). 1013-1044. doi:10.4208/cicp.020611.090312a
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