Certain types of Runge-Kutta-type formulas and reproduction of orbits of linear systems

Abstract

Sanz-Serna derived a condition for a Runge-Kutta method to give a symplectic mapping when applied to an arbitrary Hamiltonian system. The methods with the condition have a very interesting property even if the system to be integrated is not Hamiltonian. This paper discusses numerical reproduction of the phase portrait of a linear systemwhen using a Runge-Kutta method which satisfies one condition in addition to that of Sanz-Serna. Not only is such a method Astable, but its absolute stability region also coincides completely with the left half plane of a complex plane, and, accordingly, either convergence or divergence of atrue solution is reproduced numerically whatever the value of astepsize. Furthermore, it is shown that every elliptic closed orbit is drawn accurately by the numerical method as long as the calculation error is ignored. An explicit Runge-Kutta method, which usually is employed, does not have these properties. Finally, a four-dimensional linear system is designated which admits all elliptic orbits, and convergent and divergent solutions. It is illustrated that its phase portrait is reproduced well by using the implicit midpoint method—one of the methods with the forementioned conditions.