Superconvergence and time evolution of discontinuous Galerkin finite element solutions
Journal of Computational Physics
Journal of Scientific Computing
Journal of Scientific Computing
Journal of Scientific Computing
SIAM Journal on Numerical Analysis
A Fully-Discrete Local Discontinuous Galerkin Method for Convection-Dominated Sobolev Equation
Journal of Scientific Computing
Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws
Journal of Computational and Applied Mathematics
Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws
Journal of Scientific Computing
Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods
Journal of Scientific Computing
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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 $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$. However, error estimates for higher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes.