Analysis of the dynamics of local error control via a piecewise continuous residual

D.J. Higham, A.M. Stuart

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems.Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation-dissipative, contractive and gradient systems are analysed in this way.
LanguageEnglish
Pages44-57
Number of pages13
JournalBIT Numerical Mathematics
Volume38
Issue number1
Publication statusPublished - Mar 1998

Fingerprint

Piecewise continuous
Error Control
Interpolants
Tolerance
Differential equation
Gradient System
Dissipative Equations
Differential equations
MATLAB
Control Strategy
Ordinary differential equation
Ordinary differential equations
Numerical Solution
Tend
Analogue
Converge
Numerical Simulation
Zero
Computer simulation

Keywords

  • error control
  • continuous interpolants
  • dissipativity
  • contractivity
  • gradient systems
  • computer science
  • software engineering
  • mathematics

Cite this

@article{be57363d5b334f30959465e0c4c4f087,
title = "Analysis of the dynamics of local error control via a piecewise continuous residual",
abstract = "Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems.Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation-dissipative, contractive and gradient systems are analysed in this way.",
keywords = "error control, continuous interpolants, dissipativity, contractivity, gradient systems, computer science, software engineering, mathematics",
author = "D.J. Higham and A.M. Stuart",
year = "1998",
month = "3",
language = "English",
volume = "38",
pages = "44--57",
journal = "BIT Numerical Mathematics",
issn = "0006-3835",
number = "1",

}

Analysis of the dynamics of local error control via a piecewise continuous residual. / Higham, D.J.; Stuart, A.M.

In: BIT Numerical Mathematics, Vol. 38, No. 1, 03.1998, p. 44-57.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analysis of the dynamics of local error control via a piecewise continuous residual

AU - Higham, D.J.

AU - Stuart, A.M.

PY - 1998/3

Y1 - 1998/3

N2 - Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems.Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation-dissipative, contractive and gradient systems are analysed in this way.

AB - Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems.Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation-dissipative, contractive and gradient systems are analysed in this way.

KW - error control

KW - continuous interpolants

KW - dissipativity

KW - contractivity

KW - gradient systems

KW - computer science

KW - software engineering

KW - mathematics

UR - http://www.springerlink.com/openurl.asp?genre=issue&eissn=1572-9125&volume=38&issue=1

UR - http://www.csc.kth.se/BIT/

M3 - Article

VL - 38

SP - 44

EP - 57

JO - BIT Numerical Mathematics

T2 - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 1

ER -