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Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 2086–2108 | Cite as

Convergence Analysis of the Finite Difference ADI Scheme for Variable Coefficient Parabolic Problems with Nonzero Dirichlet Boundary Conditions

  • B. BialeckiEmail author
  • M. DryjaEmail author
  • R. I. FernandesEmail author
Article
  • 9 Downloads

Abstract

Since the invention by Peaceman and Rachford, more than 60 years ago, of the well celebrated ADI finite difference scheme for parabolic initial-boundary problems on rectangular regions, many papers have been concerned with prescribing the boundary values for the intermediate approximations at half time levels in the case of nonzero Dirichlet boundary conditions. In the present paper, for variable coefficient parabolic problems and time-stepsize sufficiently small, we prove second order accuracy in the discrete \({{L}^{2}}\) norm of the ADI finite difference scheme in which the intermediate approximations do not involve the so called “perturbation term”. As a byproduct of our stability analysis we also show that, for variable coefficients and time-stepsize sufficiently small, the ADI scheme with the perturbation term converges with order two in the discrete \({{H}^{1}}\) norm. Our convergence results generalize previous results obtained for the heat equation.

Keywords:

parabolic equation finite difference ADI convergence analysis. 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Statistics, Colorado School of MinesGoldenUSA
  2. 2.Department of Informatics, Vistula UniversityWarsawPoland
  3. 3.Department of Applied Mathematics and Sciences, Petroleum Institute, Khalifa University of Science and TechnologyAbu DhabiUnited Arab Emirates

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