Numerical Methods

Projects with a strong component in the development of numerical methods and numerical analysis. Currently we focus on structure preservation and multiscale time integration.

Structure Preservation

Many different physical phenomena like wave-propagation, gas dynamics, acoustics, shallow-water flows can be accurately modeled by hyperbolic partial differential equations(PDEs). The striking feature of nonlinear hyperbolic PDEs is that smooth initial data can lead in finite time to discontinuous, multivalued solutions. The presence of discontinuous, in fact multi-valued, solutions poses a substantial difficulty to the mathematical treatment thereof. Although these problems were already studied since the days of Riemann, Rankine, and Hugoniot in the 19th century, it took until the second half of the last century to observe that the thermodynamic entropy provides the sought guidance to resolve this loss of uniqueness.

Since then, preserving structural properties of the physical models in the discretized numerical routines is one of the main challenges in current computational physics.

Multiscale Time Integration

The seminal CFL condition due to Courant, Friedrichs and Lewy constrains the admissible timestep proportional to the gridsize and characteristics speeds of the model. In particular, for non-uniform meshes refined in regions of interest much smaller timesteps than in non-refined regions are required.

To address this efficiently, multiscale time integration methods are investigated.