Central local discontinuous galerkin methods on overlapping cells for diffusion equations

Yingjie Liu; Chi-Wang Shu; Eitan Tadmor; Mengping Zhang

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 6, page 1009-1032
  • ISSN: 0764-583X

Abstract

top
In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.

How to cite

top

Liu, Yingjie, et al. "Central local discontinuous galerkin methods on overlapping cells for diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1009-1032. <http://eudml.org/doc/276346>.

@article{Liu2011,
abstract = { In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees. },
author = {Liu, Yingjie, Shu, Chi-Wang, Tadmor, Eitan, Zhang, Mengping},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Central discontinuous Galerkin method; local discontinuous Galerkin method; overlapping cells; diffusion equation; heat equation; stability; error estimate; central discontinuous Galerkin method; overlapping cells; numerical experiments},
language = {eng},
month = {6},
number = {6},
pages = {1009-1032},
publisher = {EDP Sciences},
title = {Central local discontinuous galerkin methods on overlapping cells for diffusion equations},
url = {http://eudml.org/doc/276346},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Liu, Yingjie
AU - Shu, Chi-Wang
AU - Tadmor, Eitan
AU - Zhang, Mengping
TI - Central local discontinuous galerkin methods on overlapping cells for diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 6
SP - 1009
EP - 1032
AB - In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.
LA - eng
KW - Central discontinuous Galerkin method; local discontinuous Galerkin method; overlapping cells; diffusion equation; heat equation; stability; error estimate; central discontinuous Galerkin method; overlapping cells; numerical experiments
UR - http://eudml.org/doc/276346
ER -

References

top
  1. R. Bellman, The stability of solutions of linear differential equations. Duke Math. J.10 (1943) 643–647.  
  2. P. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland (1975).  
  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.  
  4. B. Cockburn and C.-W. Shu, Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput.16 (2001) 173–261.  
  5. B. Cockburn, B. Dong, J. Guzman, M. Restelli and R. Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput.31 (2009) 3827–3846.  
  6. 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.  
  7. Y.-J. Liu, C.-W. Shu, E. Tadmor and M. Zhang, L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: M2AN42 (2008) 593–607.  
  8. B. van Leer and S. Nomura, Discontinuous Galerkin for diffusion, in Proceedings of 17th AIAA Computational Fluid Dynamics Conference (2005) 2005–5108.  
  9. M. van Raalte and B. van Leer, Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Comm. Comput. Phys.5 (2009) 683–693.  
  10. 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.  
  11. M. Zhang and C.-W. Shu, An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput. Fluids34 (2005) 581–592.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.