Lack of dissipativity is not symplecticness

We show that, when numerically integrating Hamiltonian problems, nondissipative numerical methods do not in general share the advantages possessed by symplectic integrators. Here a numerical method is called nondissipative if, when applied with a small stepsize to the test equationdy/dt = iλy, λ real, has amplification factors of unit modulus. We construct a fourth order, nondissipative, explicit Runge-Kutta-Nyström procedure with small error constants. Numerical experiments show that this scheme does not perform efficiently in the numerical integration of Hamiltonian problems.

[1]  Wojciech Rozmus,et al.  A symplectic integration algorithm for separable Hamiltonian functions , 1990 .

[2]  Robert D. Skeel,et al.  Explicit canonical methods for Hamiltonian systems , 1992 .

[3]  Y. Suris.  Some properties of methods for the numerical integration of systems of the form x¨ = f ( x ) , 1989 .

[4]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[5]  Y. Suris,et al.  The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems x¨ = - 6 U/ 6 x , 1989 .

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

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

[8]  Mari Paz Calvo,et al.  High-Order Symplectic Runge-Kutta-Nyström Methods , 1993, SIAM J. Sci. Comput..

[9]  Robert D. Skeel,et al.  An explicit Runge-Kutta-Nystro¨m method is canonical if and only if its adjoint is explicit , 1992 .

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

[11]  R. Ruth,et al.  Fourth-order symplectic integration , 1990 .

[12]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

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

[14]  J. M. Sanz-Serna,et al.  Order conditions for canonical Runge-Kutta-Nyström methods , 1992 .