Geometrisk numerisk integrering av differentialekvationer

Abstract

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.