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

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Abstract

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 ∈ [a,b], subject to u(a) = α and u^′(a) = β. By carefully choosing suitable numerical fluxes and employing a special projection, we established optimal error estimates in the L²-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 ≥ 2 toward a special projection of the exact solution. Additionally, we proved that the p-degree UWDG solution and its derivative were 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 ≥ 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.

Keywords

ultra-weak discontinuous Galerkin method / nonlinear initial-value problems / optimal error estimate / superconvergence error analysis

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