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

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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

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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

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