Improved accuracy for the approximate factorization of parabolic equations

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.

Original languageEnglish
Pages (from-to)23-36
Number of pages14
JournalComputing (Vienna/New York)
Volume86
Issue number1
DOIs
Publication statusPublished - Sep 2009

Fingerprint

Approximate Factorization
Factorization
Parabolic Equation
Alternating Direction
Linear multistep Methods
Implicit Scheme
Factorization Method
Higher Order
Term
Approximation

Keywords

  • ADI method
  • Approximate factorization
  • Linear multistep method
  • Stability

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Computer Science Applications
  • Software
  • Computational Mathematics
  • Numerical Analysis
  • Theoretical Computer Science

Cite this

Improved accuracy for the approximate factorization of parabolic equations. / Karaa, S.

In: Computing (Vienna/New York), Vol. 86, No. 1, 09.2009, p. 23-36.

Research output: Contribution to journalArticle

@article{aaac342ac04643dbbcecb6e08d2ba410,
title = "Improved accuracy for the approximate factorization of parabolic equations",
abstract = "A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.",
keywords = "ADI method, Approximate factorization, Linear multistep method, Stability",
author = "S. Karaa",
year = "2009",
month = "9",
doi = "10.1007/s00607-009-0063-6",
language = "English",
volume = "86",
pages = "23--36",
journal = "Computing (Vienna/New York)",
issn = "0010-485X",
publisher = "Springer Wien",
number = "1",

}

TY - JOUR

T1 - Improved accuracy for the approximate factorization of parabolic equations

AU - Karaa, S.

PY - 2009/9

Y1 - 2009/9

N2 - A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.

AB - A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.

KW - ADI method

KW - Approximate factorization

KW - Linear multistep method

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=69949087765&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69949087765&partnerID=8YFLogxK

U2 - 10.1007/s00607-009-0063-6

DO - 10.1007/s00607-009-0063-6

M3 - Article

AN - SCOPUS:69949087765

VL - 86

SP - 23

EP - 36

JO - Computing (Vienna/New York)

JF - Computing (Vienna/New York)

SN - 0010-485X

IS - 1

ER -