Optimal energy-conserving local discontinuous Galerkin method for the one-dimensional sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose and analyze a high-order and energy-conserving local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We prove the energy-conserving property and the L-2 stability for the semi-discrete LDG method. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L-2-norm for the semi-discrete formulation. In particular, we identify a special numerical flux and a particular projection of the initial conditions for the LDG scheme for which the L-2-norm of the solution and its spatial derivative are of order p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. Several numerical results are presented to validate the theoretical analyze of the proposed algorithm. It appears that similar conclusions are valid for the two-dimensional case.