Long-term stability of multi-value methods for ordinary di erential equations
Raffaele D'Ambrosio and Ernst Hairer
Abstract.
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^{p+4}exp(h^2Lt), where p
is the order of the method, and L depends on the problem and on the
coeffcients of the method. This is confirmed by numerical experiments.
Key Words.
Multi-value methods, general linear methods, backward error analysis,
modulated Fourier expansion, parasitic components,
Hamiltonian systems, long-term integration.