Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems
HTML articles powered by AMS MathViewer
- by W. H. Hundsdorfer and J. G. Verwer PDF
- Math. Comp. 53 (1989), 81-101 Request permission
Abstract:
In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initial-boundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems where the semidiscrete system, obtained after discretization in space, satisfies a one-sided Lipschitz condition with a constant independent of the grid spacing. For such problems, unconditional stability and convergence results will be derived.References
- K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 774402
- Jim Douglas Jr. and James E. Gunn, A general formulation of alternating direction methods. I. Parabolic and hyperbolic problems, Numer. Math. 6 (1964), 428–453. MR 176622, DOI 10.1007/BF01386093
- Jim Douglas Jr. and H. H. Rachford Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956), 421–439. MR 84194, DOI 10.1090/S0002-9947-1956-0084194-4
- E. G. D′jakonov, Difference schemes with splitting operator for higher-dimensional non-stationary problems, Ž. Vyčisl. Mat i Mat. Fiz. 2 (1962), 549–568 (Russian). MR 203955
- G. Fairweather and A. R. Mitchell, A new computational procedure for $\textrm {A.D.I.}$ methods, SIAM J. Numer. Anal. 4 (1967), 163–170. MR 218027, DOI 10.1137/0704016
- A. R. Gourlay, Splitting methods for time dependent partial differential equations, The state of the art in numerical analysis (Proc. Conf., Univ. York, Heslington, 1976) Academic Press, London, 1977, pp. 757–796. MR 0451759
- A. R. Gourlay and J. Ll. Morris, The extrapolation of first order methods for parabolic partial differential equations. II, SIAM J. Numer. Anal. 17 (1980), no. 5, 641–655. MR 588750, DOI 10.1137/0717054
- A. R. Gourlay and Andrew R. Mitchell, The equivalence of certain alternating direction and locally one-dimensional difference methods, SIAM J. Numer. Anal. 6 (1969), 37–46. MR 250492, DOI 10.1137/0706004
- P. J. van der Houwen and J. G. Verwer, One-step splitting methods for semidiscrete parabolic equations, Computing 22 (1979), no. 4, 291–309 (English, with German summary). MR 620058, DOI 10.1007/BF02265311 P. J. van der Houwen & B. P. Sommeijer, Improving the Stability of Predictor-Corrector Methods by Residue Smoothing, Report NM-R8707, Centre for Mathematics and Computer Science, Amsterdam, 1987.
- J. F. B. M. Kraaijevanger, $B$-convergence of the implicit midpoint rule and the trapezoidal rule, BIT 25 (1985), no. 4, 652–666. MR 811280, DOI 10.1007/BF01936143
- Peter Lancaster and Miron Tismenetsky, The theory of matrices, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR 792300
- D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41. MR 71874 R. D. Richtmeyer & K. W. Morton, Difference Methods for Initial-Value Problems, Interscience, New York, 1967.
- J. M. Sanz-Serna, J. G. Verwer, and W. H. Hundsdorfer, Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1987), no. 4, 405–418. MR 875165, DOI 10.1007/BF01396661
- J. M. Sanz-Serna and J. G. Verwer, Stability and convergence at the PDE/stiff ODE interface, Appl. Numer. Math. 5 (1989), no. 1-2, 117–132. Recent theoretical results in numerical ordinary differential equations. MR 979551, DOI 10.1016/0168-9274(89)90028-7
- B. P. Sommeijer, P. J. van der Houwen, and J. G. Verwer, On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations, Internat. J. Numer. Methods Engrg. 17 (1981), no. 3, 335–346. MR 608685, DOI 10.1002/nme.1620170304
- J. G. Verwer, Contractivity of locally one-dimensional splitting methods, Numer. Math. 44 (1984), no. 2, 247–259. MR 753957, DOI 10.1007/BF01410109
- J. G. Verwer and J. M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations, Computing 33 (1984), no. 3-4, 297–313. MR 773930, DOI 10.1007/BF02242274
- J. G. Verwer and H. B. de Vries, Global extrapolation of a first order splitting method, SIAM J. Sci. Statist. Comput. 6 (1985), no. 3, 771–780. MR 791198, DOI 10.1137/0906052
- J. G. Verwer, Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines, Numerical analysis (Dundee, 1985) Pitman Res. Notes Math. Ser., vol. 140, Longman Sci. Tech., Harlow, 1986, pp. 220–237. MR 873112
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 81-101
- MSC: Primary 65N40; Secondary 65M20
- DOI: https://doi.org/10.1090/S0025-5718-1989-0969489-7
- MathSciNet review: 969489