𝐵-convergence properties of multistep Runge-Kutta methods

Li, Shou Fu

doi:10.1090/S0025-5718-1994-1201071-0

$B$-convergence properties of multistep Runge-Kutta methods
HTML articles powered by AMS MathViewer

by Shou Fu Li PDF

Math. Comp. 62 (1994), 565-575 Request permission

Abstract:

By using the theory of B-convergence for general linear methods to the special case of multistep Runge-Kutta methods, a series of B-convergence results for multistep Runge-Kutta methods is obtained, and it is proved that the family of algebraically stable r-step s-stage multistep Runge-Kutta methods with parameters ${\alpha _1},{\alpha _2}, \ldots ,{\alpha _r}$ presented by Burrage in 1987 is optimally Bconvergent of order at least s, and B-convergent of order $s + 1$, provided that $r \geq s$ and ${\alpha _j} > 0,j = 1,2, \ldots ,r$. Furthermore, this family of methods is optimally B-convergent of order $s + 1$ if some other additional conditions are satisfied.

References

K. Burrage, High order algebraically stable multistep Runge-Kutta methods, SIAM J. Numer. Anal. 24 (1987), no. 1, 106–115. MR 874738, DOI 10.1137/0724009
K. Burrage and W. H. Hundsdorfer, The order of $B$-convergence of algebraically stable Runge-Kutta methods, BIT 27 (1987), no. 1, 62–71. MR 874861, DOI 10.1007/BF01937355
G. J. Cooper, On the existence of solutions for algebraically stable Runge-Kutta methods, IMA J. Numer. Anal. 6 (1986), no. 3, 325–330. MR 967672, DOI 10.1093/imanum/6.3.325
M. Crouzeix, W. H. Hundsdorfer, and M. N. Spijker, On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT 23 (1983), no. 1, 84–91. MR 689606, DOI 10.1007/BF01937328
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
Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no. 3, 497–514. MR 787573, DOI 10.1137/0722030
Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no. 3, 497–514. MR 787573, DOI 10.1137/0722030
W. H. Hundsdorfer and M. N. Spijker, On the algebraic equations in implicit Runge-Kutta methods, SIAM J. Numer. Anal. 24 (1987), no. 3, 583–594. MR 888751, DOI 10.1137/0724040
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, DOI 10.1007/BF01936143
Shou Fu Li, $B$-convergence of general linear methods, BAIL V (Shanghai, 1988) Boole Press Conf. Ser., vol. 12, Boole, Dún Laoghaire, 1988, pp. 203–208. MR 990265
Shou Fu Li, Stability and $B$-convergence of general linear methods, Proceedings of the 3rd International Congress on Computational and Applied Mathematics (Leuven, 1988), 1989, pp. 281–296. MR 1038851, DOI 10.1016/0377-0427(89)90340-3

On the existence and uniqueness of solutions for implicit multivalue multiderivative methods

Ivar Lie and Syvert P. Nørsett, Superconvergence for multistep collocation, Math. Comp. 52 (1989), no. 185, 65–79. MR 971403, DOI 10.1090/S0025-5718-1989-0971403-5