Optimal error estimates and superconvergence analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems for ODEs

The primary focus of this study was to analyze the convergence and superconvergence properties of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems (IVPs) for ordinary differential equations (ODEs) of the form $ u''+(g(x, u))' = f(x, u), \ x\in [a, b], $ subject to $ u(a) = \alpha $ and $ u'(a) = \beta $. By carefully choosing suitable numerical fluxes and employing a special projection, we established optimal error estimates in the $ L^2 $-norm. The order of convergence was proved to be $ p+1 $, when utilizing piecewise polynomials of degree at most $ p $. We further proved that the UWDG solution was superconvergent of order $ p+2 $ for $ p\geq 2 $ toward a special projection of the exact solution. Additionally, we proved that the $ p $-degree UWDG solution and its derivative were $ \mathcal{O}(h^{2p}) $ superconvergent at the end of each step. Our proofs were valid for arbitrary uniform or non-uniform partitions of the domain using piecewise polynomials with degree $ p\geq 2 $. Finally, several numerical examples were provided to validate all theoretical results. It is worth noting that the proposed UWDG method offers a significant advantage for second-order differential equations, as it can be applied directly without introducing auxiliary variables or reformulating the equation as a first-order system. This advantage reduces memory and computational costs.