TY - JOUR
T1 - High-Order Energy Stable Numerical Schemes for a Nonlinear Variational Wave Equation Modeling Nematic Liquid Crystals in Two Dimensions
AU - P. Aursand & U. Koley
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 20
EP - 47
PY - 2016
DA - 2016/01
SN - 14
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/408.html
KW - Nonlinear variational wave equation, energy preserving scheme, energy stable scheme, discontinuous Galerkin method, higher order scheme.
AB - <p style="text-align: justify;">We consider a nonlinear variational wave equation that models the dynamics of the
director field in nematic liquid crystals with high molecular rotational inertia. Being derived from
an energy principle, energy stability is an intrinsic property of solutions to this model. For the two-dimensional
case, we design numerical schemes based on the discontinuous Galerkin framework
that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments
are performed verifying the scheme&#39;s energy stability, order of convergence and computational
efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian
scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose
regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge
to different solutions.</p>