Abstract
Symmetric methods of this chapter and symplectic methods of the next chapter play a central role in the geometric integration of differential equations. We discuss reversible differential equations and reversible maps, and we explain how symmetric integrators are related to them. We study symmetric Runge-Kutta and composition methods, and we show how standard approaches for solving differential equations on manifolds can be symmetrized. A theoretical explanation of the excellent longtime behaviour of symmetric methods applied to reversible differential equations will be given in Chap. XI.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
The study of symmetric methods has its origin in the development of extrapolation methods (Gragg 1965, Stetter 1973), because the global error admits an asymptotic expansion in even powers of h. The notion of time-reversible methods is more common in the Computational Physics literature (Buneman 1967).
For irreducible Runge-Kutta methods, the condition (2.4) is also necessary for symmetry (after a suitable permutation of the stages).
The authors are grateful to S. Blanes for this reference.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hairer, E., Wanner, G., Lubich, C. (2002). Symmetric Integration and Reversibility. In: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05018-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-05018-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-05020-0
Online ISBN: 978-3-662-05018-7
eBook Packages: Springer Book Archive