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

2006/01/01 by Qiang Zhang, Chi‐Wang Shu · 64 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computational Fluid Dynamics and Aerodynamics #Conservation law #Courant–Friedrichs–Lewy condition #Discontinuous Galerkin method #Discretization #Finite element method #Mathematical analysis #Mathematics #Nonlinear system #Numerical analysis #Numerical methods for differential equations #Physics #Piecewise #Piecewise linear function #Polynomial #Runge–Kutta methods

paper · doi:10.1137/040620382

published in SIAM Journal on Numerical Analysis 44(4), 1703-1720 (Society for Industrial and Applied Mathematics)

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27

Abstract

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the ℙ^k (piecewise polynomial) finite element is used. When k=1 (piecewise linear finite element), the error estimate is obtained under the usual CFL condition \dt≤ β h for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for ℙ^k finite elements with k>1 are obtained under a more restrictive CFL condition.

Citations

• Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar 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

• Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws
• A priori error estimates of Adams-Bashforth discontinuous Galerkin\n methods for scalar nonlinear conservation laws
• Notes on error estimates for the standard Galerkin-finite element method for the Shallow Water equations
• Error Estimates of Runge-Kutta Discontinuous Galerkin Methods for the Vlasov-Maxwell System
• Numerical analysis of an H(div)-conforming divergence-free DG method with a second-order explicit Runge-Kutta scheme for incompressible flows
• Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods. [europepmc]

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