Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u '' = f(x, u). Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order p + 1 in the L-2-norm for the solution, when piecewise polynomials of degree at most p are used. We use these results to prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p >= 3 towards a special projection of the exact solution. We further prove that the p-degree UWDG solution and its derivative are O(h(2p)) superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p >= 2. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.