Skip to main content
Log in

A general formulation of alternating direction methods

Part I. Parabolic and hyperbolic problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Birkhoff, G., andR. S. Varga: Implicit alternating direction methods. Trans. Amer. Math. Soc.92, 13–24 (1959).

    Google Scholar 

  2. Brian, P. L. T.: A finite-difference method of high-order accuracy for the solution of three-dimensional heat conduction problems. A. I. Ch. E. J.7, 367–370 (1961).

    Google Scholar 

  3. Douglas, J.: On the numerical integration ofU xx +U yy =U t by implicit methods. J. Soc. Ind. Appl. Math.3, 42–65 (1955).

    Google Scholar 

  4. —: Alternating direction methods for three space variables. Numer. Math.4, 41–63 (1961).

    Google Scholar 

  5. —: On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations. J. Soc. Ind. Appl. Math.4, 20–37 (1956).

    Google Scholar 

  6. Douglas, J.: A survey of numerical methods for parabolic differential equations. Advances in Computers, vol. II,F. L. Alt (editor), Academic Press 1961, pp. 1–54.

  7. —, andJ. E. Gunn: Two high-order correct difference analogues for the equation of multidimensional heat flow. Math. of Comp.17, 71–80 (1963).

    Google Scholar 

  8. ——: Alternating direction methods for parabolic systems inm-space variables. J. Assn. for Comp. Machinery9, 450–456 (1962).

    Google Scholar 

  9. —, andB. F. Jones jr.: On predictor-corrector methods for nonlinear parabolic equations. J. Soc. Ind. Appl. Math.11, 195–204 (1963).

    Google Scholar 

  10. —, andH. H. Rachford jr.: On the numerical solution of the heat conduction problems in two and three space variables. Trans. of the Amer. Math. Soc.82, 421–439 (1956).

    Google Scholar 

  11. Forsythe, G. E., andW. R. Wasow: Finite Difference Methods for Partial Differential Equations. New York: John Wiley and Sons, Inc. 1960.

    Google Scholar 

  12. Guilinger, W.: Private communication.

  13. Konovalov, A. N.: The method of fractional steps for solving the Cauchy problem for the multi-dimensional wave equation. Dokl. Akad. Nauk147, 25–27 (1962).

    Google Scholar 

  14. Lax, P. D., andR. D. Richtmyer: Survey of stability of linear finite difference equations. Comm. Pure Appl. Math.9, 267–293 (1956).

    Google Scholar 

  15. Lees, M.: A priori estimates for the solutions of difference approximations to parabolic differential equations. Duke Math. J.27, 297–311 (1960).

    Google Scholar 

  16. —: Alternating direction and semi-explicit difference methods for parabolic differential equations. Numer. Math.3, 398–412 (1961).

    Google Scholar 

  17. —: Alternating direction methods for hyperbolic differential equations. J. Soc. Ind. Math.10, 610–616 (1960).

    Google Scholar 

  18. Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Amer. Math. Soc., Providence, 1949.

    Google Scholar 

  19. Peaceman, D. W., andH. H. Rachford jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math.3, 28–41 (1955).

    Google Scholar 

  20. Samarskii, A. A.: Locally one-dimensional difference schemes on non-uniform grids. Ž. Vyčisl. Mat. i Mat. Fiz.3, 431–466 (1963).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Douglas, J., Gunn, J.E. A general formulation of alternating direction methods. Numer. Math. 6, 428–453 (1964). https://doi.org/10.1007/BF01386093

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01386093

Keywords

Navigation