Canonical integration methods for Hamiltonian dynamical systems

Hamiltonian systems possess dynamics (e.g., preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research in this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this thesis, we explore possibilities for the construction of integrators that preserve the property of being canonical. We characterize explicit methods as compositions of 1-stage methods. We find coefficients for methods having minimum work per step. We study in more detail the relationship between canonical, symmetric and order conditions. We also analyze linearized stability. We perform numerical experiments to investigate the importance of being canonical. Examining the solution orbits we discover that numerical integrators possess the same characteristics as the flow of the original system. However, for a variety of test problems we demonstrate equally good results for integrations that merely have the right linearized stability property. We also perform numerical experiments to investigate the improvement that higher-order integrators have over lower-order integrators. This seems to be an interesting research problem, since lower-order methods are currently widely in use in real practical simulations. To direct interest to the application of higher-order methods, serious numerical experiments have to be made to find out their efficiency over lower-order methods. We show that for the two-body problem, the error growth in time for variable step size canonical methods is not in general quadratic. The error growth in time in the case of second-order integrators, canonical or non-canonical turns out to be approximately linear.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Okunbor, Daniel Irowa

Contributors dc:contributor

• Skeel, Robert D.

• Mathematics
• Computer Science

Statement dc:rights

• Copyright 1993 Okunbor, Daniel Irowa

Language dc:language

eng

Identifier

AAI9314922
(UMI)AAI9314922

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/21597