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A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Compressible fluids and gas dynamics, general Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  21 June 2017

Huiqiang Yue ,
Jian Cheng  and
Tiegang Liu
Huiqiang Yue*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
Jian Cheng*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
Tiegang Liu*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
*
*Corresponding author. Email addresses:yuehq@buaa.edu.cn (H. Yue), chengjian@buaa.edu.cn (J. Cheng), liutg@buaa.edu.cn (T. Liu)
*Corresponding author. Email addresses:yuehq@buaa.edu.cn (H. Yue), chengjian@buaa.edu.cn (J. Cheng), liutg@buaa.edu.cn (T. Liu)
*Corresponding author. Email addresses:yuehq@buaa.edu.cn (H. Yue), chengjian@buaa.edu.cn (J. Cheng), liutg@buaa.edu.cn (T. Liu)
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Abstract

In this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

Keywords

Compressible Navier-Stokes equationsdiscontinuous Galerkin methodcompressible viscous flow

MSC classification

Secondary: 35Q30: Navier-Stokes equations 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 375 - 392
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Chi-Wang Shu

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