Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



On optimal high-order in time approximations for the Korteweg-de Vries equation

Authors: Ohannes Karakashian and William McKinney
Journal: Math. Comp. 55 (1990), 473-496
MSC: Primary 65N30; Secondary 35Q53
MathSciNet review: 1035935
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the well-known order reduction phenomenon affecting implicit Runge-Kutta methods does not occur when approximating periodic solutions of the Korteweg-de Vries equation.

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

  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1970. MR 0450957 (56:9247)
  • [2] G. A. Baker, V. A. Dougalis, and O. A. Karakashian, Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp. 40 (1983), 419-433. MR 689464 (84f:65072)
  • [3] J. L. Bona, J. A. Dougalis, and O. A. Karakashian, Conservative high order schemes for the generalized Korteweg-de Vries equation (submitted).
  • [4] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555-604. MR 0385355 (52:6219)
  • [5] J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 60-64. MR 0159424 (28:2641)
  • [6] M. Crouzeix, Sur l'approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, Thèse, Université Paris VI, 1975.
  • [7] K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, North-Holland, Amsterdam, 1984. MR 774402 (86g:65003)
  • [8] V. A. Dougalis and O. A. Karakashian, On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp. 45 (1985), 329-345. MR 804927 (86m:65118)
  • [9] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [10] O. A. Karakashian, On Runge-Kutta methods for parabolic problems with time-dependent coefficients, Math. Comp. 47 (1986), 77-101. MR 842124 (87i:65161)
  • [11] O. A. Karakashian and W. Rust, On the parallel implementation of implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 9 (1988), 1085-1090. MR 963856 (89h:65112)
  • [12] V. Thomée and B. Wendroff, Convergence estimates for Galerkin methods for variable coefficient initial value problems, SIAM J. Numer. Anal. 11 (1974), 1059-1068. MR 0371088 (51:7309)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 35Q53

Retrieve articles in all journals with MSC: 65N30, 35Q53

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society