Geometric Numerical Integration

The lecture will be recorded and uploaded here (no live stream), together with revised lecture notes.

Problem sheets with solution proposals will be given. Assistance to the problem sheets is provided in the ILIAS discussion forum. You may submit your solutions. Programming exercises will be supplied with a screen recording that shows the purpose of the program.

This is not a final list. If you have suggestions and need some other form of assistance, please let us know in the forum.

The numerical simulation of time-dependent processes in science and technology often leads to the problem to solve a system of ordinary differential equations with a suitable method. In many applications it can be shown that the exact flow exhibits certain qualitative or "geometric'' properties. For example, it is well-known that the flow of a Hamiltonian system is symplectic, and that the energy remains constant along the solution although the solution itself changes in time.

When the solution or the flow is approximated by a numerical integrator, it is desirable to preserve these geometric properties, because reproducing the correct qualitative behavior is important in many applications. It turns out, however, that only selected methods respect the geometric properties of the dynamics. These methods are called geometric numerical integrators. In this lecture we will investigate
- why certain methods are (or are not) geometric numerical integrators,
- how to construct geometric numerical integrators,
- which properties are conserved, and in which sense.

Prof. Dr. Tobias Jahnke (Lecture), MSc Benny Stein (Problem class)

3+1

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1550284