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

Karakashian, Ohannes; McKinney, William

doi:10.1090/S0025-5718-1990-1035935-4

On optimal high-order in time approximations for the Korteweg-de Vries equation
HTML articles powered by AMS MathViewer

by Ohannes Karakashian and William McKinney PDF

Math. Comp. 55 (1990), 473-496 Request permission

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

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Garth A. Baker, Vassilios A. Dougalis, and Ohannes A. Karakashian, Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp. 40 (1983), no. 162, 419–433. MR 689464, DOI 10.1090/S0025-5718-1983-0689464-4

Conservative high order schemes for the generalized Korteweg-de Vries equation

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), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9

Sur l’approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta

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
Vassilios A. Dougalis and Ohannes A. Karakashian, On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp. 45 (1985), no. 172, 329–345. MR 804927, DOI 10.1090/S0025-5718-1985-0804927-8
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Ohannes A. Karakashian, On Runge-Kutta methods for parabolic problems with time-dependent coefficients, Math. Comp. 47 (1986), no. 175, 77–101. MR 842124, DOI 10.1090/S0025-5718-1986-0842124-1
Ohannes A. Karakashian and William Rust, On the parallel implementation of implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 9 (1988), no. 6, 1085–1090. MR 963856, DOI 10.1137/0909074
Vidar Thomée and Burton Wendroff, Convergence estimates for Galerkin methods for variable coefficient initial value problems, SIAM J. Numer. Anal. 11 (1974), 1059–1068. MR 371088, DOI 10.1137/0711081