Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge-Kutta Discontinuous Galerkin Method for Scalar Conservation Laws

  • Authors:

  • Qiang Zhang;Chi-Wang Shu

  • Affiliations:

  • qzh@nju.edu.cn;shu@dam.brown.edu

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 2010

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Abstract

In this paper we present an analysis of the Runge-Kutta discontinuous Galerkin method for solving scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta method. We use an energy technique to prove the $\mathrm{L}^2$-norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree $k\geq1$ under the standard temporal-spatial CFL condition $\tau\leq\gamma h$, where $h$ and $\tau$ are the element length and time step, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$.