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We consider a nonlinear variational wave equation that models the dynamics ofthe director field in nematic liquid crystals with high molecular rotationalinertia. Being derived from an energy principle, energy stability is anintrinsic property of solutions to this model. For the two-dimensional case, wedesign numerical schemes based on the discontinuous Galerkin framework thateither conserve or dissipate a discrete version of the energy.  Extensive numerical experiments are performed verifying the scheme's energystability, order of convergence and computational efficiency. The numericalsolutions are compared to those of a simpler first-order Hamiltonian scheme. Weprovide numerical evidence that solutions of the 2D variational wave equationloose regularity in finite time. After that occurs, dissipative andconservative schemes appear to converge to different solutions.

High-order energy stable numerical schemes for a nonlinear variational wave equation modeling nematic liquid crystals in two dimensions