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 $u^{″}+(g(x,u){)}^{\prime }=f(x,u),x\in [a,b],$ subject to $u(a)=\alpha$ and $u^{\prime }(a)=\beta$. 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 $p\ge 2$ 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 $p\ge 2$. 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.