Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems

Authors:
W. H. Hundsdorfer and J. G. Verwer

Journal:
Math. Comp. **53** (1989), 81-101

MSC:
Primary 65N40; Secondary 65M20

MathSciNet review:
969489

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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****[2]**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**0176622****[3]**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**0084194**, 10.1090/S0002-9947-1956-0084194-4**[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**0203955****[5]**G. Fairweather and A. R. Mitchell,*A new computational procedure for 𝐴.𝐷.𝐼. methods*, SIAM J. Numer. Anal.**4**(1967), 163–170. MR**0218027****[6]**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****[7]**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**, 10.1137/0717054**[8]**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**0250492****[9]**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**, 10.1007/BF02265311**[10]**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.**[11]**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**, 10.1007/BF01936143**[12]**Peter Lancaster and Miron Tismenetsky,*The theory of matrices*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR**792300****[13]**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**0071874****[14]**R. D. Richtmeyer & K. W. Morton,*Difference Methods for Initial-Value Problems*, Interscience, New York, 1967.**[15]**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**, 10.1007/BF01396661**[16]**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**, 10.1016/0168-9274(89)90028-7**[17]**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**, 10.1002/nme.1620170304**[18]**J. G. Verwer,*Contractivity of locally one-dimensional splitting methods*, Numer. Math.**44**(1984), no. 2, 247–259. MR**753957**, 10.1007/BF01410109**[19]**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**, 10.1007/BF02242274**[20]**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**, 10.1137/0906052**[21]**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**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N40,
65M20

Retrieve articles in all journals with MSC: 65N40, 65M20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969489-7

Keywords:
Numerical analysis,
time-dependent PDE's,
alternating direction implicit methods,
Peaceman-Rachford method,
method of lines,
stability,
error bounds

Article copyright:
© Copyright 1989
American Mathematical Society