Abstract
The aim of this work is to understand the possible role in the long-term integration of conservative systems of “G-symplectic” methods. It comes out of a collaboration with Dr Adrian Hill of the University of Bath, United Kingdom, and Dr Yousaf Habib of the National University of Science and Technology, Pakistan. Although symplectic behaviour, or the exact conservation of quadratic invariants, for irreducible methods of this type, is not possible [3], there is a G-generalization, similar to the generalization introduced by Dahlquist [6] in the study of non-linear dissipative methods. Specific issues in this research include the role of time-reversal symmetry in conservative integration, the exacerbation of parasitic effects and the construction and implementation of specific methods of increasingly high orders.
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.
Similar content being viewed by others
References
Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal. 16, 46–57 (1979)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester and New York (2008)
Butcher, J.C., Hewitt, L.L.: The existence of symplectic general linear methods. Numerical Algorithms 51, 77–84 (2009)
Butcher, J.C., Habib, Y., Hill, A.T., Norton, T.: Control of parasitism in G-symplectic methods (in preparation, 2012)
Butcher, J.C., Hill, A.T.: Symmetric general linear methods (in preparation, 2012)
Dahlquist, G.: Error analysis for a class of methods for stiff non–linear initial value problems. Lecture Notes in Mathematics, vol. 506, pp. 60–72 (1976)
Cooper, G.J.: Stability of Runge–Kutta methods for trajectory problems. IMA J. Numer. Anal. 7, 1–13 (1987)
Hammer, P.C., Hollingsworth, J.W.: Trapezoidal methods of approximating solutions of differential equations. MTAC 9, 92–96 (1955)
Heun, K.: Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen. Z. Math. Phys. 45, 23–38
Kutta, W.: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901)
Runge, C.: Über die numerische Auflösung von Differentialgleichungen. Math. Ann. 46, 167–178 (1895)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Butcher, J.C. (2013). Dealing with Parasitic Behaviour in G-Symplectic Integrators. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33221-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-33221-0_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33220-3
Online ISBN: 978-3-642-33221-0
eBook Packages: EngineeringEngineering (R0)