An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation

Abstract

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 "damping" 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.