An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation (Communications in Computational Physics, 2018)
We design and numerically validate a local discontinuous Galerkin (LDG)method to compute solutions to the initial value problem for a nonlinear variationalwave equation originally proposed to model liquid crystals. For the semi-discrete LDGformulation with a class of alternating numerical fluxes, the energy conserving propertyis verified. A dissipative scheme is also introduced by locally imposing somenumerical "damping" in the scheme so as to suppress some numerical oscillations nearsolution singularities. Extensive numerical experiments are presented to validate andillustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ isnumerically obtained when using alternating numerical fluxes. When using the centralnumerical flux, only sub-optimal convergence is observed for polynomials of odddegree. Numerical simulations with long time integration indicate that the energyconserving property is crucial for accurately capturing the underlying wave shapes.
Keywords: Discontinuous Galerkin method; variational wave equation; energy conservation.