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Extrapolation of symplectic methods for Hamiltonian problems

✍ Scribed by R.P.K. Chan; A. Murua

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
144 KB
Volume
34
Category
Article
ISSN
0168-9274

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✦ Synopsis

We consider two modes of the extrapolation of symplectic and symmetric Runge-Kutta and related integrators over long time-intervals applied with constant stepsize. In the passive mode, we compute two solution sequences with stepsizes h and h/2 independently and perform extrapolation whenever output is required. In the active mode, we extrapolate at every step and propagate the extrapolated solution. We study and compare in detail both modes of extrapolation applied to the simple harmonic oscillator. We show that passive extrapolation will improve the accuracy of the numerical solution over the whole integration interval even though it destroys the linear error growth of the basic method and that active extrapolation exhibits linear error growth for the harmonic oscillator and, in general, yields higher accuracy than passive extrapolation.

The error growth of integrations over long time-intervals is also studied in a more general setting. We obtain asymptotic error formulas for the periodic case and for integrable Hamiltonian systems where linear error growth has been established in the study of Calvo and Hairer [Appl. Numer. Math. 18 (1995) 95-105].

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Multi-Symplectic Runge–Kutta Collocation
Multi-Symplectic Runge–Kutta Collocation Methods for Hamiltonian Wave Equations
✍ Sebastian Reich πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 403 KB

A number of conservative PDEs, like various wave equations, allow for a multisymplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss-Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numer