Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On monotonicity and boundedness properties of linear multistep methods


Authors: Willem Hundsdorfer and Steven J. Ruuth
Journal: Math. Comp. 75 (2006), 655-672
MSC (2000): Primary 65L06, 65M06, 65M20
DOI: https://doi.org/10.1090/S0025-5718-05-01794-1
Published electronically: November 17, 2005
MathSciNet review: 2196985
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also consider contractivity and stability in arbitrary norms.


References [Enhancements On Off] (What's this?)

  • 1. M. Tawarmalani, N.V. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Nonconvex Optimization and Its Applications 65, Kluwer, 2002. MR 1961018 (2004a:90001)
  • 2. G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems, Procs. Dundee Conf. 1975, Lecture Notes in Math. 506, G.A. Watson (ed.), Springer, 1976, pp. 60-74. MR 0448898 (56:7203)
  • 3. L. Ferracina, M.N. Spijker, Stepsize restrictions for total-variation-boundedness in general Runge-Kutta procedures. Appl. Numer. Math. 53 (2005), pp. 265-279. MR 2128526
  • 4. S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability preserving high-order time discretization methods, SIAM Review 42 (2001), pp. 89-112. MR 1854647 (2002f:65132)
  • 5. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I - Nonstiff Problems, Second edition, Springer Series in Comput. Math. 8, Springer, 1993. MR 1227985 (94c:65005)
  • 6. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems, Second edition, Springer Series in Comput. Math. 14, Springer, 1996. MR 1439506 (97m:65007
  • 7. W. Hundsdorfer, S.J. Ruuth and R.J. Spiteri, Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal. 41 (2003), pp. 605-623. MR 2004190 (2004g:65093)
  • 8. W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Comput. Math. 33, Springer, 2003. MR 2002152 (2004g:65001)
  • 9. R. Jeltsch, O. Nevanlinna, Stability of explicit time discretizations for solving initial value problems, Numer. Math. 37 (1981), pp. 61-91. MR 0615892 (82g:65042)
  • 10. H.W.J. Lenferink, Contractivity preserving explicit linear multistep methods, Numer. Math. 55 (1989), pp. 213-223. MR 0987386 (90f:65058)
  • 11. H.W.J. Lenferink, Contractivity preserving implicit linear multistep methods, Math. Comp. 56 (1991), pp. 177-199. MR 1052098 (91i:65129)
  • 12. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. MR 1925043 (2003h:65001)
  • 13. S.J. Ruuth, W. Hundsdorfer, High-order linear multistep methods with general monotonicity and boundedness properties. To appear in J. Comp. Phys., 2005.
  • 14. J. Sand, Circle contractive linear multistep methods, BIT 26 (1986), pp. 114-122. MR 0833836 (87h:65124)
  • 15. C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp. 49 (1987), pp. 105-121. MR 0890256 (89b:65208)
  • 16. C.-W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comp. 9 (1988), pp. 1073-1084. MR 0963855 (90a:65196)
  • 17. M.N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math. 42 (1983), pp. 271-290. MR 0723625 (85b:65067)
  • 18. R. Vanselov, Nonlinear stability behaviour of linear multistep methods, BIT 23 (1983), pp. 388-396. MR 0705005 (84k:65090)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65L06, 65M06, 65M20

Retrieve articles in all journals with MSC (2000): 65L06, 65M06, 65M20


Additional Information

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

Steven J. Ruuth
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada
Email: sruuth@sfu.ca

DOI: https://doi.org/10.1090/S0025-5718-05-01794-1
Keywords: Multistep schemes, monotonicity, boundedness, TVD, TVB, contractivity, stability
Received by editor(s): March 10, 2004
Received by editor(s) in revised form: January 6, 2005
Published electronically: November 17, 2005
Additional Notes: The work of the second author was partially supported by a grant from NSERC Canada.
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society