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Trapezoidal and midpoint splittings
for initial-boundary value problems


Author: Willem Hundsdorfer
Journal: Math. Comp. 67 (1998), 1047-1062
MSC (1991): Primary 65M06, 65M12, 65M20
DOI: https://doi.org/10.1090/S0025-5718-98-00984-3
MathSciNet review: 1484899
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider various multi-component splittings based on the trapezoidal rule and the implicit midpoint rule. It will be shown that an important requirement on such methods is internal stability. The methods will be applied to initial-boundary value problems. Along with a theoretical analysis, some numerical test results will be presented.


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  • 1. R.M. Beam, R.F. Warming, An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. J. Comp. Phys. 22, pp. 87-110 (1976). MR 56:13673
  • 2. P. Brenner, M. Crouzeix, V. Thomée, Single step methods for inhomogeneous linear differential equations in Banach space. RAIRO Anal. Numer. 16, pp. 5-26 (1982). MR 83d:65268
  • 3. R. \v{C}iegis, K. Ki\v{s}kis, On the stability of LOD difference schemes with respect to boundary conditions. Lithuanian Academy of Sciences, Informatica 5, pp. 297-323 (1994). MR 96j:65085
  • 4. E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I - Nonstiff Problems. Springer Verlag, Berlin, 1987. MR 87m:65005
  • 5. P.J. van der Houwen, J.G. Verwer, One-step splitting methods for semi-discrete parabolic equations. Computing 22, pp. 291-309 (1979). MR 83e:65148
  • 6. W. Hundsdorfer, Unconditional convergence of some Crank-Nicolson LOD methods for initial-boundary value problems. Math. Comp. 58, pp. 35-53 (1992). MR 92e:65124
  • 7. W. Hundsdorfer, A note on stability of the Douglas splitting method. CWI Report, 1996.
  • 8. W. Hundsdorfer, J.G. Verwer, Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math. Comp. 53, pp. 81-101 (1989). MR 90h:65195
  • 9. J.F.B.M Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule. BIT 25, pp. 652-666 (1985). MR 87c:65096
  • 10. R.J. LeVeque, Intermediate boundary conditions for LOD, ADI and approximate factorization methods. ICASE Report 85-21, Langley Research Center, 1985.
  • 11. Ch. Lubich, A. Ostermann, Interior estimates for time discretization of parabolic equations. Appl. Num. Math. 18, pp. 241-251 (1995). MR 96f:65124
  • 12. G.I. Marchuk, Splitting and alternating direction methods. Handbook of Numerical Analysis 1 (P.G. Ciarlet. J.L. Lions, eds.), North-Holland, Amsterdam, pp. 197-462, 1990. CMP 90:08
  • 13. A.R. Mitchell, D.F. Griffiths, The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, Chichester, 1980. MR 82a:65002
  • 14. J.G. Verwer, J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations. Computing 33, pp. 297-313 (1984). MR 86k:65085
  • 15. N.N. Yanenko, The Method of Fractional Steps. Springer Verlag, Berlin, 1971. MR 46:6613

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Additional Information

Willem Hundsdorfer
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: w.hundsdorfer@cwi.nl

DOI: https://doi.org/10.1090/S0025-5718-98-00984-3
Keywords: Numerical analysis, initial-boundary value problems, splitting methods
Received by editor(s): July 29, 1996
Additional Notes: Part of the research for this paper was performed during a visit at the University of Coimbra (Portugal) for the EU/HCM project CRHX-0930407.
Article copyright: © Copyright 1998 American Mathematical Society

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