Energy preservation in separable Hamiltonian systems by splitting schemes
Abstract
Splitting and composition schemes for the numerical integration of separable Hamiltonian problems, in general, fail to yield conservation of generic nonlinear Hamiltonians. In the proposed approach we compose, at each step, symplectic maps which depend on a parameter chosen in order to have methods which minimize the error on the Hamiltonian function. Second and fourth order symmetric methods are proposed which are, in the particular case of the linear oscillator, symplectic, energypreserving and explicit. On the nonlinear oscillator, the energy preservation and the simplecticity are not obtained exactly but retained at some extent. The proposed methods are semiexplicit in the sense that they require, as additional computational effort, the search for a zero of a scalar function with respect to a scalar variable.
 Publication:

Numerical Analysis and Applied Mathematics ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics
 Pub Date:
 September 2012
 DOI:
 10.1063/1.4756367
 Bibcode:
 2012AIPC.1479.1204P
 Keywords:

 differential equations;
 error analysis;
 integration;
 numerical analysis;
 02.30.Cj;
 02.30.Hq;
 02.30.Jr;
 02.60.Jh;
 02.60.Lj;
 Measure and integration;
 Ordinary differential equations;
 Partial differential equations;
 Numerical differentiation and integration;
 Ordinary and partial differential equations;
 boundary value problems