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

2004/01/01 by Qiang Zhang, Chi-Wang Shu, Chi‐Wang Shu · 206 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 #Fluid Dynamics and Turbulent Flows #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Monotone polygon #Nonlinear system #Numerical analysis #Order (exchange) #Physics #Piecewise #Quantum mechanics #Runge–Kutta methods #Scalar (mathematics)

paper · doi:10.1137/s0036142902404182

published in SIAM Journal on Numerical Analysis 42(2), 641-666 (Society for Industrial and Applied Mathematics)

openalex publication_date 2004/01/01 · 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 scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the ℙ¹ (piecewise linear) elements are obtained under the usual CFL condition τ≤ γ h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and τ are the maximum element lengths and time steps, respectively, and the positive constant γ is independent of h and τ. However, error estimates forhigher order ℙ^k(k≥ 2) elements need a more restrictive time step τ≤ γ h^4/3. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition τ≤γ h for the ℙ^k elements of degree k≥ 2. Error estimates of O(h^k+1/2+τ²) are obtained for general monotone numerical fluxes, and optimal error estimates of O(h^k+1+τ²) are obtained for upwind numerical fluxes.

Citations

• Mixed finite elements in ?3
• 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

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