Geometric Integration Methods for Materials and Fluids

Atomistic models of materials, i.e. molecular dynamics, result in large systems of Newtonian equations of motion that evolve over a wide range of time and length scales. Computational techniques are required to reproduce the statistical mechanics of such systems in a reliable and efficient manner. In that respect, the concept of geometric integration, in particular symplectic integration, has been proven to be a fundamental guiding principle.

In the first part of my talk, I will give an introduction to symplectic integration and computational statistical mechanics. This will be followed by a discussion of some of the challenges arising from biomolecular modelling and the need for multi-scale methods in space and time.

From a mathematical point of view Hamiltonian dynamics, symplectic geometry and asymptotic expansions play a key role in the design and analysis of numerical methods. I will discuss this in the context of backward error analysis.

In the final part of my talk, these ideas will be extended to PDEs and multi-symplectic geometry. In particular, the concept of geometric integration has recently started to find its way into geophysical fluid dynamics and numerical weather prediction.