no. 4
L 2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 593-607.

We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.

DOI : 10.1051/m2an:2008018
Classification : 65M60
Mots clés : central discontinuous Galerkin method, discontinuous Galerkin method, linear hyperbolic equation, stability, error estimate 
@article{M2AN_2008__42_4_593_0,
     author = {Liu, Yingjie and Shu, Chi-Wang and Tadmor, Eitan and Zhang, Mengping},
     title = {$L^2$ stability analysis of the central discontinuous {Galerkin} method and a comparison between the central and regular discontinuous {Galerkin} methods},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {593--607},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {4},
     year = {2008},
     doi = {10.1051/m2an:2008018},
     mrnumber = {2437775},
     zbl = {1152.65095},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008018/}
}
TY  - JOUR
AU  - Liu, Yingjie
AU  - Shu, Chi-Wang
AU  - Tadmor, Eitan
AU  - Zhang, Mengping
TI  - $L^2$ stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 593
EP  - 607
VL  - 42
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008018/
DO  - 10.1051/m2an:2008018
LA  - en
ID  - M2AN_2008__42_4_593_0
ER  - 
%0 Journal Article
%A Liu, Yingjie
%A Shu, Chi-Wang
%A Tadmor, Eitan
%A Zhang, Mengping
%T $L^2$ stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 593-607
%V 42
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008018/
%R 10.1051/m2an:2008018
%G en
%F M2AN_2008__42_4_593_0
Liu, Yingjie; Shu, Chi-Wang; Tadmor, Eitan; Zhang, Mengping. $L^2$ stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 593-607. doi : 10.1051/m2an:2008018. http://www.numdam.org/articles/10.1051/m2an:2008018/

[1] P. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland (1975). | MR | Zbl

[2] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1989) 411-435. | MR | Zbl

[3] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463. | MR | Zbl

[4] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173-261. | MR | Zbl

[5] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89-112. | MR | Zbl

[6] G.-S. Jiang and C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62 (1994) 531-538. | MR | Zbl

[7] Y.J. Liu, Central schemes on overlapping cells. J. Comput. Phys. 209 (2005) 82-104. | MR | Zbl

[8] Y.J. Liu, C.-W. Shu, E. Tadmor and M. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45 (2007) 2442-2467. | MR | Zbl

[9] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR | Zbl

[10] J. Qiu, B.C. Khoo and C.-W. Shu, A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes. J. Comput. Phys. 212 (2006) 540-565. | MR | Zbl

[11] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439-471. | MR | Zbl

[12] M. Zhang and C.-W. Shu, An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13 (2003) 395-413. | MR | Zbl

[13] M. Zhang and C.-W. Shu, An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids 34 (2005) 581-592. | Zbl

Cité par Sources :