### 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 language | English |
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Pages (from-to) | 23-36 |

Number of pages | 14 |

Journal | Computing (Vienna/New York) |

Volume | 86 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 2009 |

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### 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.

Research output: Contribution to journal › Article

*Computing (Vienna/New York)*, vol. 86, no. 1, pp. 23-36. https://doi.org/10.1007/s00607-009-0063-6

}

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

VL - 86

SP - 23

EP - 36

JO - Computing (Vienna/New York)

JF - Computing (Vienna/New York)

SN - 0010-485X

IS - 1

ER -