The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls. (English) Zbl 0947.65133

Ascher, Uri M.; Reich, Sebastian

SIAM J. Sci. Comput. 21, No. 3, 1045-1065 (1999).

The authors study the suitability of the (implicit) midpoint rule for the integration of highly oscillatory Hamiltonian systems as they often appear in molecular dynamics simulations. They show that one must impose rather tight restrictions on the stepsize in order to be sure that no misleading results are obtained. In particular, they show that even the errors in slowly varying quantities (adiabatic invariants) like the energy may exhibit an unexpected growth, if a too large stepsize is used. To make things worse, the numerical results may give no indications that something goes wrong. The analysis is based on the detailed examination of some simple examples (mainly stiff oscillators, single and coupled). All theoretical results are tested in numerical computations.

Reviewer: Werner M.Seiler (Mannheim)

MSC:

65P10	Numerical methods for Hamiltonian systems including symplectic integrators
65L20	Stability and convergence of numerical methods for ordinary differential equations
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
37M15	Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65L50	Mesh generation, refinement, and adaptive methods for ordinary differential equations

Keywords:

implicit midpoint rule; Hamiltonian systems; highly oscillatory systems; numerical stability; molecular dynamics; numerical examples; stepsize control; stiff oscillators