Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension

In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG 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. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.