Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 235, Issue -, Pages 458-485Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2012.10.020
Keywords
Discontinuous Galerkin method; Local discontinuous Galerkin method; Fourier approach; Dispersion and dissipation error; Eigen-structure; Superconvergence
Funding
- ICERM
- Air Force Office of Scientific Computing YIP [FA9550-12-0318]
- NSF
- University of Houston
- [DMS-0914852]
- [DMS-1217008]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1217008] Funding Source: National Science Foundation
Ask authors/readers for more resources
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 Delta x. 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. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available