Energy conserving, Liouville, and symplectic integrators (Q1902391)

In the last few years most research in the numerical solution of ordinary differential equations has been addressed to the development of methods adapted to special problems. In particular, a complete theory of symplectic methods for Hamiltonian systems has been constructed [see e.g. \textit{J. M. Sanz-Serna} and \textit{M. P. Calvo}, Numerical Hamiltonian problems (1994; Zbl 0816.65042)]. The present author explores, from a numerical point of view, the possibilities of some numerical methods satisfying a weaker requirement than symplecticity. Thus, in the first part he introduces a particular technique to conserve the energy in a Hamiltonian system by rescaling the velocity at each step so that the energy remains constant along the numerical integration. Some numerical results are presented to show that for the Henon-Heiles problem this technique destroys some important dynamical properties of the phase space. Secondly, the author explores two-stage Runge-Kutta-Nyström methods conserving the volume in the phase-space. Since for problems with dimension \(\geq 2\) this property is a weaker requirement than symplecticity, the author compares a second-order preserving volume method with the Stormer-Verlet symplectic method. Numerical experiments for the two-body problem and a pseudospectral discretization of the sine- Gordon equation show that both methods have a similar behavior for these problems.