Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations

In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L²-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h^2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise P^p polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.