A hybrid Padé ADI scheme of higher-order for convection-diffusion problems

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

A high-order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection-diffusion problems. The scheme employs standard high-order Padé approximations for spatial first and second derivatives in the convection-diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87-110) are applied for the time integration. The approximate factorization imposes a second-order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher-order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC-based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high-order ADI schemes for solving unsteady convection-diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given.

Original languageEnglish
Pages (from-to)532-548
Number of pages17
JournalInternational Journal for Numerical Methods in Fluids
Volume64
Issue number5
DOIs
Publication statusPublished - Oct 2010

Fingerprint

Alternating Direction
Convection-diffusion Problems
Implicit Scheme
Factorization
Higher Order
Approximate Factorization
Linear multistep Methods
Reynolds number
Derivatives
Linear systems
Time Integration
Higher Order Approximation
Convection-diffusion Equation
Cell
Time Discretization
Term
Second derivative
Convection
Costs
Linear Systems

Keywords

  • ADI method
  • Approximate factorization
  • High-order compact scheme
  • Linear multistep method
  • Pade approximations
  • Unsteady convection-diffusion equation

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Mechanics
  • Applied Mathematics
  • Mechanical Engineering
  • Mechanics of Materials

Cite this

A hybrid Padé ADI scheme of higher-order for convection-diffusion problems. / Karaa, Samir.

In: International Journal for Numerical Methods in Fluids, Vol. 64, No. 5, 10.2010, p. 532-548.

Research output: Contribution to journalArticle

@article{f390a531920549e285c2c00e09681da1,
title = "A hybrid Pad{\'e} ADI scheme of higher-order for convection-diffusion problems",
abstract = "A high-order Pad{\'e} alternating direction implicit (ADI) scheme is proposed for solving unsteady convection-diffusion problems. The scheme employs standard high-order Pad{\'e} approximations for spatial first and second derivatives in the convection-diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87-110) are applied for the time integration. The approximate factorization imposes a second-order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher-order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC-based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high-order ADI schemes for solving unsteady convection-diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given.",
keywords = "ADI method, Approximate factorization, High-order compact scheme, Linear multistep method, Pade approximations, Unsteady convection-diffusion equation",
author = "Samir Karaa",
year = "2010",
month = "10",
doi = "10.1002/fld.2160",
language = "English",
volume = "64",
pages = "532--548",
journal = "International Journal for Numerical Methods in Fluids",
issn = "0271-2091",
publisher = "John Wiley and Sons Ltd",
number = "5",

}

TY - JOUR

T1 - A hybrid Padé ADI scheme of higher-order for convection-diffusion problems

AU - Karaa, Samir

PY - 2010/10

Y1 - 2010/10

N2 - A high-order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection-diffusion problems. The scheme employs standard high-order Padé approximations for spatial first and second derivatives in the convection-diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87-110) are applied for the time integration. The approximate factorization imposes a second-order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher-order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC-based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high-order ADI schemes for solving unsteady convection-diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given.

AB - A high-order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection-diffusion problems. The scheme employs standard high-order Padé approximations for spatial first and second derivatives in the convection-diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87-110) are applied for the time integration. The approximate factorization imposes a second-order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher-order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC-based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high-order ADI schemes for solving unsteady convection-diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given.

KW - ADI method

KW - Approximate factorization

KW - High-order compact scheme

KW - Linear multistep method

KW - Pade approximations

KW - Unsteady convection-diffusion equation

UR - http://www.scopus.com/inward/record.url?scp=77956931063&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956931063&partnerID=8YFLogxK

U2 - 10.1002/fld.2160

DO - 10.1002/fld.2160

M3 - Article

AN - SCOPUS:77956931063

VL - 64

SP - 532

EP - 548

JO - International Journal for Numerical Methods in Fluids

JF - International Journal for Numerical Methods in Fluids

SN - 0271-2091

IS - 5

ER -