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

2010/01/01 by Qiang Zhang, Chi-Wang Shu, Chi‐Wang Shu · 4 citations
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #Advanced Numerical Methods in Computational Mathematics #Numerical methods for differential equations

paper · doi:10.1137/090771363

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 L²-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≥1 under the standard temporal-spatial CFL condition τ≤γ h, where h and τ are the element length and time step, respectively, and the positive constant γ is independent of h and τ.

Citations

• Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws
• Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws
• The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V
• The Finite Element Method for Elliptic Problems
• Efficient implementation of essentially non-oscillatory shock-capturing schemes

Cited by

• Necessity of numerical smoothness
• Enforcing strong stability of explicit Runge--Kutta methods with superviscosity
• Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems
• Stationary discrete shock profiles for scalar conservation laws with a\n discontinuous Galerkin method

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• Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws
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