Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Numerical methods
- 3 Hamiltonian mechanics
- 4 Geometric integrators
- 5 The modified equations
- 6 Higher-order methods
- 7 Constrained mechanical systems
- 8 Rigid body dynamics
- 9 Adaptive geometric integrators
- 10 Highly oscillatory problems
- 11 Molecular dynamics
- 12 Hamiltonian PDEs
- References
- Index
4 - Geometric integrators
Published online by Cambridge University Press: 04 December 2009
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Numerical methods
- 3 Hamiltonian mechanics
- 4 Geometric integrators
- 5 The modified equations
- 6 Higher-order methods
- 7 Constrained mechanical systems
- 8 Rigid body dynamics
- 9 Adaptive geometric integrators
- 10 Highly oscillatory problems
- 11 Molecular dynamics
- 12 Hamiltonian PDEs
- References
- Index
Summary
In Chapter 2, we introduced the concept of a numerical integrator as a mapping which approximates the flow-map of a given system of differential equations. We have also seen a few instances of how such integrators behave, demonstrating concepts such as convergence and order of accuracy. We observed that the typical picture is a locally accurate approximation that gradually drifts further from the true trajectory (see Fig. 2.3, Fig. 2.5 and the left panel of Fig. 2.7); the rate of drift can be reduced by reducing the stepsize (and thereby also increasing the amount of computational work), but the qualitative picture does not change in any significant way.
What stands out as remarkable, therefore, is the behavior, illustrated in the right panel of Fig. 2.7, of the Euler-B method, which retains bounded trajectories when applied to the harmonic oscillator. In Chapter 2, we provided an explanation for this in the form of a linear stability analysis showing that certain methods, including Störmer–Verlet and Euler-B, have eigenvalues on the unit circle when applied to the harmonic oscillator (or any other oscillatory linear system), if the stepsize is below some threshold value. The Euler-B and Störmer-Verlet methods (among others) possess a strong asymptotic stability property for linear systems.
It is interesting to note that a related long-term stability property extends to nonlinear models. If we apply, for example, the Störmer–Verlet methods to the Lennard-Jones oscillator, we obtain the results illustrated in Fig. 4.1 (compare with Fig. 2.3 and Fig. 2.5).
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- Simulating Hamiltonian Dynamics , pp. 70 - 104Publisher: Cambridge University PressPrint publication year: 2005