This paper focuses on the analysis of convergence and superconvergence properties of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order Volterra integro-differential equations. The key idea for deriving optimal error estimates in the \(L^2\) -norm is the use of appropriately chosen numerical fluxes together with a specially designed projection. When piecewise polynomials of degree \(p \ge 2\) are employed, the proposed UWDG method is shown to achieve the optimal convergence order of \(p+1\) . Moreover, we prove that the UWDG solution exhibits superconvergence of order \(p+2\) toward a suitable projection of the exact solution. In addition, both the p-degree UWDG solution and its derivative are shown to be \(\mathcal {O}(h^{2p})\) superconvergent at the end of each time step. These theoretical results hold for arbitrary regular meshes and piecewise polynomial spaces of degree \(p \ge 2\) . Finally, several numerical experiments are presented to confirm the theoretical findings. Notably, the proposed UWDG method offers a significant advantage over standard discontinuous Galerkin methods for first-order systems, as it can be applied directly without introducing auxiliary variables or reformulating the problem as a larger system, thereby reducing both memory requirements and computational cost.
