Efficient implementation of stable Richardson Extrapolation algorithms

I. Faragó, Gnes Havasi, Zahari Zlatev

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated),the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used,the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process,experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved andthe computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.

Original languageEnglish
Pages (from-to)2309-2325
Number of pages17
JournalComputers and Mathematics with Applications
Volume60
Issue number8
DOIs
Publication statusPublished - Oct 2010

Fingerprint

Richardson Extrapolation
Efficient Implementation
Extrapolation
Numerical methods
Numerical Methods
System of Ordinary Differential Equations
Ordinary differential equations
Euler's formula
Atmospheric chemistry
Trapezoidal Rule
Stiff Systems
Combined Method
Convergence of numerical methods
Algebraic Equation
Chemistry
Experiment
Linear systems
Computational Cost
Nonlinear systems
Approximate Solution

Keywords

  • Atmospheric chemistry scheme
  • Backward Euler Formula
  • Richardson Extrapolation
  • Sparse matrix technique
  • Stability
  • Trapezoidal rule

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Modelling and Simulation
  • Computational Mathematics

Cite this

Efficient implementation of stable Richardson Extrapolation algorithms. / Faragó, I.; Havasi, Gnes; Zlatev, Zahari.

In: Computers and Mathematics with Applications, Vol. 60, No. 8, 10.2010, p. 2309-2325.

Research output: Contribution to journalArticle

@article{61f68ad0b6254e2cb7bdb9fba57eaf74,
title = "Efficient implementation of stable Richardson Extrapolation algorithms",
abstract = "Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated),the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used,the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process,experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved andthe computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.",
keywords = "Atmospheric chemistry scheme, Backward Euler Formula, Richardson Extrapolation, Sparse matrix technique, Stability, Trapezoidal rule",
author = "I. Farag{\'o} and Gnes Havasi and Zahari Zlatev",
year = "2010",
month = "10",
doi = "10.1016/j.camwa.2010.08.025",
language = "English",
volume = "60",
pages = "2309--2325",
journal = "Computers and Mathematics with Applications",
issn = "0898-1221",
publisher = "Elsevier Limited",
number = "8",

}

TY - JOUR

T1 - Efficient implementation of stable Richardson Extrapolation algorithms

AU - Faragó, I.

AU - Havasi, Gnes

AU - Zlatev, Zahari

PY - 2010/10

Y1 - 2010/10

N2 - Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated),the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used,the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process,experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved andthe computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.

AB - Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated),the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used,the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process,experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved andthe computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.

KW - Atmospheric chemistry scheme

KW - Backward Euler Formula

KW - Richardson Extrapolation

KW - Sparse matrix technique

KW - Stability

KW - Trapezoidal rule

UR - http://www.scopus.com/inward/record.url?scp=77957361855&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957361855&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2010.08.025

DO - 10.1016/j.camwa.2010.08.025

M3 - Article

AN - SCOPUS:77957361855

VL - 60

SP - 2309

EP - 2325

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 8

ER -