@Article{CiCP-21-211,
author = {Waixiang Cao, Dongfang Li and Zhimin Zhang},
title = {Optimal Superconvergence of Energy Conserving Local Discontinuous Galerkin Methods for Wave Equations},
journal = {Communications in Computational Physics},
year = {2018},
volume = {21},
number = {1},
pages = {211--236},
abstract = {<p style="text-align: justify;">This paper is concerned with numerical solutions of the LDG method for
1D wave equations. Superconvergence and energy conserving properties have been
studied. We first study the superconvergence phenomenon for linear problems when
alternating fluxes are used. We prove that, under some proper initial discretization, the
numerical trace of the LDG approximation at nodes, as well as the cell average, converge
with an order 2k+1. In addition, we establish k+2-th order and k+1-th order
superconvergence rates for the function value error and the derivative error at Radau
points, respectively. As a byproduct, we prove that the LDG solution is superconvergent
with an order k+2 towards the Radau projection of the exact solution. Numerical
experiments demonstrate that in most cases, our error estimates are optimal, i.e., the
error bounds are sharp. In the second part, we propose a fully discrete numerical
scheme that conserves the discrete energy. Due to the energy conserving property, after
long time integration, our method still stays accurate when applied to nonlinear
Klein-Gordon and Sine-Gordon equations.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.120715.100516a},
url = {http://global-sci.org/intro/article_detail/cicp/11238.html}
}