Abstract
The 1976 paper of G. Dahlquist, [13], has had a wide-ranging impact on our understanding of numerical methods for the solution of stiff differential equation systems. The present paper surveys some of the work of Dahlquist in this area. It also shows how this has led to contributions by other authors. In particular, the paper contains a review of non-linear stability for Runge–Kutta and general linear methods.
Similar content being viewed by others
References
K. Burrage and J. C. Butcher, Stability criteria for implicit Runge–Kutta methods, SIAM J. Numer. Anal., 16 (1979), pp. 46–57.
K. Burrage and J. C. Butcher, Non-linear stability of a general class of differential equation methods, BIT, 20 (1980), pp. 185–203.
J. C. Butcher, On the convergence of numerical solutions of ordinary differential equations, Math. Comp., 20 (1966), pp. 1–10.
J. C. Butcher, A stability property of implicit Runge–Kutta methods, BIT, 15 (1975), pp. 358–361.
J. C. Butcher, Linear and non-linear stability for general linear methods, BIT, 27 (1987), pp. 182–189.
J. C. Butcher, The equivalence of algebraic stability and AN-stability, BIT, 27 (1987), pp. 510–533.
J. C. Butcher and A. T. Hill, Linear multistep methods as irreducible general linear methods, BIT, 46 (2006), pp. 5–19.
M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math., 32 (1979), pp. 75–82.
C. F. Curtiss and J. O. Hirschfelder, Integration of stiff equations, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), pp. 235–243.
G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956), pp. 33–53.
G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), pp. 27–43.
G. Dahlquist, On stability and error analysis for stiff non-linear problems, 1, Dept of Information Processing, Royal Institute of Technology, Stockholm, Report NA 75.08 (1975).
G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems, Proc. Numer. Anal. Conf., Dundee, Scotland, 1975, Lecture Notes Math., vol. 506, pp. 60–74, Springer, New York, 1976.
G. Dahlquist, G-stability is equivalent to A-stability, BIT, 18 (1978), pp. 384–401.
G. Dahlquist, W. Liniger, and O. Nevanlinna, Stability of two-step methods for variable integration steps, SIAM J. Numer. Anal., 20 (1983), pp. 1071–1085.
G. Dahlquist and G. Söderlind, Some problems related to stiff nonlinear differential systems, in Computing methods in Applied Sciences and Engineering V, R. Glowinski, J. L.Lions, eds., pp. 57–74, North Holland, Amsterdam, 1982.
R. Frank, J. Schneid, and C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981), pp. 753–780.
F. Lasagni, Canonical Runge–Kutta methods, Z. Angew. Math. Phys., 39 (1988), pp. 952–953.
O. Nevanlinna and F. Odeh, Multiplier techniques for linear multistep methods, Numer. Funct. Anal. Optim., 3 (1981), pp. 377–423.
J. M. Sanz-Serna, Runge–Kutta schemes for Hamiltonian systems, BIT, 28 (1988), pp. 877–883.
J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numer., 1 (1991), pp. 243–286.
Y. B. Suris, Canonical transformations generated by methods of Runge–Kutta type for the numerical integration for the system x”=-∂U/∂x, Zh. Vychisl. Mat. i Mat. Fiz., 29 (1989), pp. 202–211 (in Russian).
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Germund Dahlquist (1925–2005).
AMS subject classification (2000)
65L05, 65L06, 65L20
Rights and permissions
About this article
Cite this article
Butcher, J. Thirty years of G-stability . Bit Numer Math 46, 479–489 (2006). https://doi.org/10.1007/s10543-006-0078-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-006-0078-8