TY - JOUR
T1 - Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods
T2 - Eigen-structure analysis based on Fourier approach
AU - Guo, Wei
AU - Zhong, Xinghui
AU - Qiu, Jing Mei
N1 - Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2013/2/15
Y1 - 2013/2/15
N2 - 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: 2. k + 1 for DG method and 2. k + 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 Δ 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.
AB - 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: 2. k + 1 for DG method and 2. k + 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 Δ 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.
KW - Discontinuous Galerkin method
KW - Dispersion and dissipation error
KW - Eigen-structure
KW - Fourier approach
KW - Local discontinuous Galerkin method
KW - Superconvergence
UR - http://www.scopus.com/inward/record.url?scp=84870992688&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2012.10.020
DO - 10.1016/j.jcp.2012.10.020
M3 - Article
AN - SCOPUS:84870992688
VL - 235
SP - 458
EP - 485
JO - Journal of Computational Physics
JF - Journal of Computational Physics
SN - 0021-9991
ER -