Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.

Original languageEnglish (US)
Pages (from-to)292-313
Number of pages22
JournalJournal of Computational and Applied Mathematics
Volume333
DOIs
StatePublished - May 1 2018

Keywords

  • Error estimates
  • Local discontinuous Galerkin method
  • Projections
  • Sine-Gordon equation
  • Superconvergence

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension'. Together they form a unique fingerprint.

  • Cite this