Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

Mahboub Baccouch

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u″ = f(x,u). Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order p + 1 in the L2-norm for the solution, when piecewise polynomials of degree at most p are used. We use these results to prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards a special projection of the exact solution. We further prove that the p-degree UWDG solution and its derivative are O(h2p) superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p ≥ 2. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

Original languageEnglish (US)
Article number2250042
JournalInternational Journal of Computational Methods
Volume20
Issue number2
DOIs
StatePublished - Mar 1 2023
Externally publishedYes

Keywords

  • Second-order initial-value problems
  • a priori error estimate
  • superconvergence
  • ultra-weak discontinuous Galerkin method

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Computational Mathematics

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