Geometrisk numerisk integrering av differentialekvationer
Typ
Examensarbete för kandidatexamen
Program
Publicerad
2022
Författare
Sahlin, Erik
Kyhn, Georg
Karlander, William
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
The report analyses the symplectic Euler method and the St¨ormer-Verlet method applied to
Hamiltonian systems. The methods are numerically applied to three different hamiltonian
systems and are compared to other numeric methods such as the explicit Euler method and
the implicit Euler method. The first Hamiltonian system that is studied is the ideal pendulum.
The second is a Hamiltonian system consisting of two celestial bodies. The third Hamiltonian
system is a molecular dynamics problem consisting of two atoms. Examples are given to show
important properties of Hamiltonian systems that later are defined and proved for general
cases. These important properties includes invariants and symplectic maps. Proofs of the
symplectic numerical methods conservation of first integrals and proofs of which methods are
symplectic are performed. Which problems that have symplectic solutions are also shown.
Kepler’s problem is one of the examples brought to light. Here two celestial bodies are
modeled with one orbiting the other. The four numerical methods (the explicit, implicit and
symplectic Euler method, and the St¨ormer-Verlet method) are applied and deviations from
the exact solution are examined. Specifically the orbit, the deviation of energy in the system,
angular momentum and position created by the numerical methods.
The report shows that all Hamiltonian problems have a symplectc map as well as showing
that the symplectic Euler method and St¨ormer-Verlet method are symplectic. The report
also shows that symplectic numerical methods conserve energy approximatly when applied to
Hamiltonian problems. This is shown by carrying out simulations which show numerically
that the symplectic Euler method and St¨ormer-Verlet method conserve energy within a small
inteval over exponential time. Backward error analysis is then introduced to explain this
property of symplectic numerical methods in general.