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L2 stability analysis of the central discontinuous Galerkin methodand a comparison between the central and regulardiscontinuous Galerkin methods

Published online by Cambridge University Press:  27 May 2008

Yingjie Liu ,
Chi-Wang Shu ,
Eitan Tadmor  and
Mengping Zhang
Yingjie Liu
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. yingjie@math.gatech.edu
Chi-Wang Shu
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. shu@dam.brown.edu .
Eitan Tadmor
Affiliation:
Department of Mathematics, Institute for Physical Science and Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, USA. tadmor@cscamm.umd.edu .
Mengping Zhang
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China. mpzhang@ustc.edu.cn .
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Abstract


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.

Keywords

Central discontinuous Galerkin methoddiscontinuous Galerkin methodlinear hyperbolic equationstabilityerror estimate.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 593 - 607
Copyright
© EDP Sciences, SMAI, 2008

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References

P. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland (1975).
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.
Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. CrossRef
Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173–261. CrossRef
Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. CrossRef
Jiang, G.-S. and Shu, C.-W., On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62 (1994) 531–538. CrossRef
Liu, Y.J., Central schemes on overlapping cells. J. Comput. Phys. 209 (2005) 82–104. CrossRef
Liu, Y.J., Shu, C.-W., Tadmor, E. and Zhang, M., Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45 (2007) 2442–2467. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. CrossRef
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.
Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439–471. CrossRef
Zhang, M. and Shu, C.-W., An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13 (2003) 395–413. CrossRef
Zhang, M. and Shu, C.-W., An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids 34 (2005) 581–592. CrossRef