  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'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.
