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 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.

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

“…Their work was motivated by the successful numerical experiments of Bassi and Rebay [1] for compressible Navier-Stokes equations. The LDG methods can be applied in many equations, such as KdV type equations [32,26,27,28,15,39], Camassa-Holm equations [24,35], Degasperis-Procesi equation [31], Schrödinger equations [25,23], and more nonlinear equations or system [33,24,30,34,17].…”

An energy‐based discontinuous Galerkin method for coupled elasto‐acoustic wave equations in second‐order form

“…Advantages of the proposed method are that we use the minimal number of variables required (compare with LDG [3,5,10] and HDG [4] methods) and simple conservative or upwind fluxes can be chosen to be independent of the mesh (compare with IPDG [7,9]). However, although it seems clear that the method can be adapted to any second-order linear hyperbolic system, the formulation for nonlinear problems presented in [1] is both incomplete and inconvenient.…”