The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

  • Authors:

  • Bernardo Cockburn;Chi-Wang Shu

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 1998

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Abstract

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.