Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations

Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like h exp(hLt), where p is the order of the method, and L depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.