Journal of Scientific Computing

, Volume 16, Issue 3, pp 173–261

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

  • Bernardo Cockburn
  • Chi-Wang Shu

DOI: 10.1023/A:1012873910884

Cite this article as:
Cockburn, B. & Shu, CW. Journal of Scientific Computing (2001) 16: 173. doi:10.1023/A:1012873910884


In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

discontinuous Galerkin methods non-linear conservation laws convection-diffusion equations 

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Bernardo Cockburn
    • 1
  • Chi-Wang Shu
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolis
  2. 2.Division of Applied MathematicsBrown UniversityProvidence

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