Convergence theory for initial value problems

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

Research output: Contribution to journalArticle

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)
Pages (from-to)109-144
Number of pages36
JournalMathematics in Science and Engineering
Volume213
Issue numberC
DOIs
StatePublished - 2008
Externally publishedYes

Fingerprint

Convergence Theory
Initial value problems
Initial Value Problem
Error Estimates
Long-time Asymptotics
Equivalence Theorem
Transient State
Chaotic Motion
Periodic Motion
Local Stability
Long-time Behavior
Global Stability
Ordinary differential equations
Numerical methods
Ordinary differential equation
Asymptotic Behavior
Numerical Methods
Infinity
Restriction

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)

Cite this

Convergence theory for initial value problems. / Brio, Moysey; Webb, G. M.; Zakharian, A. R.

In: Mathematics in Science and Engineering, Vol. 213, No. C, 2008, p. 109-144.

Research output: Contribution to journalArticle

Brio, Moysey ; Webb, G. M. ; Zakharian, A. R. / Convergence theory for initial value problems. In: Mathematics in Science and Engineering. 2008 ; Vol. 213, No. C. pp. 109-144.
@article{9a0bf0fac46b44fa90569b768d083b1a,
title = "Convergence theory for initial value problems",
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.",
keywords = "Consistency, Convergence, Fourier analysis, Global error estimate, Local truncation error, Order of accuracy, Resolution, Stability, Von Neumann and CFL necessary stability conditions",
author = "Moysey Brio and Webb, {G. M.} and Zakharian, {A. R.}",
year = "2008",
doi = "10.1016/S0076-5392(10)21308-5",
language = "English (US)",
volume = "213",
pages = "109--144",
journal = "Mathematics in Science and Engineering",
issn = "0076-5392",
publisher = "Elsevier",
number = "C",

}

TY - JOUR

T1 - Convergence theory for initial value problems

AU - Brio, Moysey

AU - Webb, G. M.

AU - Zakharian, A. R.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Consistency

KW - Convergence

KW - Fourier analysis

KW - Global error estimate

KW - Local truncation error

KW - Order of accuracy

KW - Resolution

KW - Stability

KW - Von Neumann and CFL necessary stability conditions

UR - http://www.scopus.com/inward/record.url?scp=77955255168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955255168&partnerID=8YFLogxK

U2 - 10.1016/S0076-5392(10)21308-5

DO - 10.1016/S0076-5392(10)21308-5

M3 - Article

AN - SCOPUS:77955255168

VL - 213

SP - 109

EP - 144

JO - Mathematics in Science and Engineering

JF - Mathematics in Science and Engineering

SN - 0076-5392

IS - C

ER -