@Article{CiCP-23-747,
author = {Nianyu Yi and Hailiang Liu},
title = {An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation},
journal = {Communications in Computational Physics},
year = {2018},
volume = {23},
number = {3},
pages = {747--772},
abstract = {<p style="text-align: justify;">We design and numerically validate a local discontinuous Galerkin (LDG)
method to compute solutions to the initial value problem for a nonlinear variational
wave equation originally proposed to model liquid crystals. For the semi-discrete LDG
formulation with a class of alternating numerical fluxes, the energy conserving property
is verified. A dissipative scheme is also introduced by locally imposing some
numerical &quot;damping&quot; in the scheme so as to suppress some numerical oscillations near
solution singularities. Extensive numerical experiments are presented to validate and
illustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ is
numerically obtained when using alternating numerical fluxes. When using the central
numerical flux, only sub-optimal convergence is observed for polynomials of odd
degree. Numerical simulations with long time integration indicate that the energy
conserving property is crucial for accurately capturing the underlying wave shapes.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2016-0189},
url = {http://global-sci.org/intro/article_detail/cicp/10546.html}
}