Convergence theory for initial value problems

M. Brio, G. M. Webb, A. R. Zakharian

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

The convergence theory for numerical methods approximating time-dependent problems parallels the theory of ordinary differential equations (ODEs) where two types of behavior are studied, namely: (1) the finite time solution and (2) the long-time asymptotic behavior where the solution either passes through an initial transient state and sets into a steady state, or evolves into a periodic or chaotic motion, or escapes to infinity. We describe the notions of consistency, stability, local and global error estimates, resolution and order of accuracy, followed by Lax-Richtmyer equivalence theorem. The rest of the chapter is devoted to practical implications of the convergence theory in terms of the resolution and error estimates together with von Neumann and CFL stability restrictions.

Original languageEnglish (US)
Title of host publicationMathematics in Science and Engineering
PublisherElsevier
Pages109-144
Number of pages36
EditionC
DOIs
StatePublished - Jan 1 2010
Externally publishedYes

Publication series

NameMathematics in Science and Engineering
NumberC
Volume213
ISSN (Print)0076-5392

Keywords

  • Consistency
  • Convergence
  • Fourier analysis
  • Global error estimate
  • Local truncation error
  • Order of accuracy
  • Resolution
  • Stability
  • Von Neumann and CFL necessary stability conditions

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Fingerprint Dive into the research topics of 'Convergence theory for initial value problems'. Together they form a unique fingerprint.

  • Cite this

    Brio, M., Webb, G. M., & Zakharian, A. R. (2010). Convergence theory for initial value problems. In Mathematics in Science and Engineering (C ed., pp. 109-144). (Mathematics in Science and Engineering; Vol. 213, No. C). Elsevier. https://doi.org/10.1016/S0076-5392(10)21308-5