285410 Simulating Hamiltonian Dynamics (V) (SoSe 2017)

This lecture will be about simulating dynamical systems, especially conservative systems such as arise in celestial mechanics and molecular models. With the rising capacity of computers and increase in the use of simulation in applications, numerical simulations techniques have become a standard tool in everything from materials modeling to bio-engineering, from atomic theory to cosmology. This is in particular true for the many physical systems that exhibit complex nonlinear dynamics or even show fundamental chaotic behavior.

After reviewing basic numerical integration schemes, we will focus on geometric integrators and numerical stability. Geometric integrators are methods that exactly (i.e. up to rounding errors) conserve qualitative properties associated to the solution of the dynamical system under study. We will apply these techniques to constrained and non-constrained Hamiltonian systems, rigid body dynamics and molecular dynamics. We will further discuss examples of highly oscillatory systems and Hamiltonian PDE's.

During the lecture we will perform and discuss numerous small numerical experiments.

Requirements for participation, required level

Basic knowledge on numerical methods as discussed in "Computer Physics" would be helpful but is not demanded.

Bibliography

- Simulating Hamiltonian Dynamics, Benedict Leimkuhler, Sebastian Reich, Cambridge University Press, 2004.

- A First Course in the Numerical Analysis of Differential Equations, Areih Iserles, Cambridge University Press, 2009.