Volume-preserving integrators have linear error growth

Abstract We present numerical evidence of linear long-term error growth in the calculation of periodic and quasi-periodic orbits of divergence-free ODEs by volume-preserving integration methods.

[1]  Uriel Frisch,et al.  Chaotic streamlines in the ABC flows , 1986, Journal of Fluid Mechanics.

[2]  G. R. W. Quispel,et al.  Numerical Integrators that Preserve Symmetries and Reversing Symmetries , 1998 .

[3]  V. Arnold Mathematical Methods of Classical Mechanics , 1974 .

[4]  D. Stoffer On the qualitative behaviour of symplectic integrators. Part II. Integrable systems , 1998 .

[5]  R. McLachlan,et al.  The accuracy of symplectic integrators , 1992 .

[6]  M. Hénon,et al.  The applicability of the third integral of motion: Some numerical experiments , 1964 .

[7]  L. Fauci,et al.  A computational model of aquatic animal locomotion , 1988 .

[8]  O. Gonzalez Time integration and discrete Hamiltonian systems , 1996 .

[9]  A. Dragt,et al.  Normal form for mirror machine Hamiltonians , 1979 .

[10]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[11]  Mari Paz Calvo,et al.  The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem , 1993, SIAM J. Sci. Comput..

[12]  Generating functions for dynamical systems with symmetries, integrals, and differential invariants , 1998 .

[13]  E. Hairer,et al.  Accurate long-term integration of dynamical systems , 1995 .

[14]  A. Neishtadt The separation of motions in systems with rapidly rotating phase , 1984 .

[15]  Robert I. McLachlan,et al.  On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods , 1995, SIAM J. Sci. Comput..

[16]  J. Marsden,et al.  Mechanical integrators derived from a discrete variational principle , 1997 .

[17]  B. Cano,et al.  Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems , 1997 .

[18]  B. Cano,et al.  Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems , 1998 .

[19]  Feng Kang,et al.  Volume-preserving algorithms for source-free dynamical systems , 1995 .

[20]  Antonella Zanna,et al.  Collocation and Relaxed Collocation for the Fer and the Magnus Expansions , 1999 .

[21]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[22]  G. Quispel,et al.  Geometric integration using discrete gradients , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[23]  G.R.W. Wuispel,et al.  Volume-preserving integrators , 1995 .

[24]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[25]  H. Yoshida Construction of higher order symplectic integrators , 1990 .

[26]  J. Butcher The Numerical Analysis of Ordinary Di erential Equa-tions , 1986 .

[27]  G. S. Turner,et al.  Discrete gradient methods for solving ODEs numerically while preserving a first integral , 1996 .

[28]  Donald Estep,et al.  The rate of error growth in Hamiltonian-conserving integrators , 1995 .