@article{Baccouch2025, 
author = {Mahboub Baccouch},
title = {Optimal error estimates and superconvergence analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems for ODEs},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {11},
pages = {6971-6997},
keywords = {ultra-weak discontinuous Galerkin method, nonlinear initial-value problems, optimal error estimate, superconvergence error analysis},
url = {https://www.sciopen.com/article/10.3934/era.2025307},
doi = {10.3934/era.2025307},
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        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  ≥  2 toward a special projection of the exact solution. Additionally, we proved that the    p-degree UWDG solution and its derivative were        O    (      h          2      p        ) 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.}
}