Optimal error estimates of the local discontinuous galerkin method for the two-dimensional sine-gordon equation on cartesian grids

The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L ² stability, and optimal L ² error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L ² -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h ^p +1) estimate.