A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation
SIAM Journal on Numerical Analysis
On a cell entropy inequality for discontinuous Galerkin methods
Mathematics of Computation
An analysis of the discontinuous Galerkin method for wave propagation problems
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Mathematics of Computation
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
Journal of Computational Physics
SIAM Journal on Scientific Computing
Journal of Scientific Computing
Journal of Scientific Computing
Superconvergence and time evolution of discontinuous Galerkin finite element solutions
Journal of Computational Physics
An analysis of the spectrum of the discontinuous Galerkin method
Applied Numerical Mathematics
Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods
Journal of Scientific Computing
Journal of Computational Physics
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Various superconvergence properties of discontinuous Galerkin (DG) and local DG (LDG) methods for linear hyperbolic and parabolic equations have been investigated in the past. Due to these superconvergence properties, DG and LDG methods have been known to provide good wave resolution properties, especially for long time integrations (Zhong and Shu, 2011) [26]. In this paper, under the assumption of uniform mesh and via Fourier approach, we observe that the error of the DG or LDG solution can be decomposed into three parts: (1) dissipation and dispersion errors of the physically relevant eigenvalue; this part of error will grow linearly in time and is of order: 2k+1 for DG method and 2k+2 for LDG method (2) projection error: there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself; this part of error will not grow in time (3) the dissipation of non-physically relevant eigenvectors; this part of error will be damped exponentially fast with respect to the spatial mesh size @Dx. Along this line, we analyze the error for a fully discrete Runge-Kutta (RK) DG scheme. A collection of numerical examples for linear equations are presented to verify our observations above. We also provide numerical examples based on non-uniform mesh, nonlinear Burgers' equation, and high-dimensional Maxwell equations to explore superconvergence properties of DG methods in a more general setting.