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A general formulation of alternating direction methods. I: Parabolic and hyperbolic problems. (English) Zbl 0141.33103

References:
[1]Birkhoff, G., andR. S. Varga: Implicit alternating direction methods. Trans. Amer. Math. Soc.92, 13-24 (1959). · doi:10.1090/S0002-9947-1959-0105814-4
[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).
[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). · Zbl 0067.35802 · doi:10.1137/0103004
[4]?: Alternating direction methods for three space variables. Numer. Math.4, 41-63 (1961). · Zbl 0104.35001 · doi:10.1007/BF01386295
[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). · Zbl 0072.14703 · doi:10.1137/0104002
[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). · doi:10.1090/S0025-5718-1963-0149676-2
[8]??: Alternating direction methods for parabolic systems inm-space variables. J. Assn. for Comp. Machinery9, 450-456 (1962).
[9]?, andB. F. Jones jr.: On predictor-corrector methods for nonlinear parabolic equations. J. Soc. Ind. Appl. Math.11, 195-204 (1963). · Zbl 0116.09104 · doi:10.1137/0111015
[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). · doi:10.1090/S0002-9947-1956-0084194-4
[11]Forsythe, G. E., andW. R. Wasow: Finite Difference Methods for Partial Differential Equations. New York: John Wiley and Sons, Inc. 1960.
[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).
[14]Lax, P. D., andR. D. Richtmyer: Survey of stability of linear finite difference equations. Comm. Pure Appl. Math.9, 267-293 (1956). · Zbl 0072.08903 · doi:10.1002/cpa.3160090206
[15]Lees, M.: A priori estimates for the solutions of difference approximations to parabolic differential equations. Duke Math. J.27, 297-311 (1960). · Zbl 0092.32803 · doi:10.1215/S0012-7094-60-02727-7
[16]?: Alternating direction and semi-explicit difference methods for parabolic differential equations. Numer. Math.3, 398-412 (1961). · Zbl 0101.34101 · doi:10.1007/BF01386038
[17]?: Alternating direction methods for hyperbolic differential equations. J. Soc. Ind. Math.10, 610-616 (1960). · Zbl 0111.29204 · doi:10.1137/0110046
[18]Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Amer. Math. Soc., Providence, 1949.
[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). · Zbl 0067.35801 · doi:10.1137/0103003
[20]Samarskii, A. A.: Locally one-dimensional difference schemes on non-uniform grids. ?. Vy?isl. Mat. i Mat. Fiz.3, 431-466 (1963).