# Research

## Numerical methods for PDEs

### State of the art

For the numerical solution of partial differential equations, a number of different schemes have been developed. Due to the stiff nature of such systems, the time step of explicit methods is restricted by a CFL condition. Thus, implicit Runge-Kutta methods or implicit multi-step methods (such as BDF) are usually employed. However, if no efficient preconditioner is available, such methods tend to be computationally expensive.

### Research

The goal is to construct, implement, and analze non-standard time integration methods (such as *splitting methods* and *exponential integrators*). Such methods are explicit and do not suffer from a CFL condition. However, their efficient implementation as well as a theoretical understanding is still an area of active research. Furthermore, we consider the application of these methods to advection-dominated PDEs (such as the Vlasov equation, the KdV equation, or magnetohydrodynamics).

**Austrian Science Fund (FWF) project**

**Splitting methods for the Vlasov-Poisson and Vlasov-Maxwell equations**

## Discontinuous Galerkin methods

### State of the art

Discontinuous Galerkin approximations have been recognized as a viable alternative to finite difference and finite elements methods. They combine the advantage of ease of handling complicated geometries with the ability to easily construct higher order methods. These properties are usually associated exclusively with the finite element and finite difference approach, respectively. Furthermore, dG methods usually provide a local approximation which can be exploited to reduce the communication overhead in parallel implementations.

### Research

Due to the discontinuous approximation, it is not always possible to extend techniques from the finite element or finite difference case in order to show convergence. Therefore, an approach has been developed that can be applied to functions with small jump discontinuities. Furthermore, the conservation properties as well as the efficient implementation on massively parallel architectures (such as graphic processing units) is of interest.

## GPU computing

### State of the art

For problems that can be parallelized, general purpose computing on graphic processing units (GPGPU) does present an alternative to more traditional multi-core systems. However, they do present additional challenges for the efficient implementation of numerical algorithms. These include: the scarcity of memory (only up to 6GB), the importance of coalesced memory access to achieve optimal performance, and the massively parallel architecture. Nevertheless, the implementation of numerical algorithms can result in significant speedups.

### Research

The implementation of matrix-free methods (due to the memory constraints) in the context of time integration schemes is of interest. We further investigate the implementation of discontinuous Galerkin methods on GPUs. Higher order dG methods have an overhead over finite difference method with respect to memory usage. However, they do require data from at most the neighboring cells and therefore the halo region is significantly decreased.

**Tiroler Wissenschaftsfonds (TWF) project**

**Exponential integrators for modern many-core architecture**

## Condensed matter physics

### State of the art

The exciton-polariton states found in certain semiconductors are a promising way of creating entangled photons. This is usually accomplished by trapping light in a (single) microcravity; in this case the strength between the laser light and the material is sufficiently increased to enable the formation of exciton-polaritons. These quasi-particles (made up of a quantum superposition of an exciton and a photon and are called exciton-polaritons) scatter and, upon decay, yield an entangled two-photon state. However, due to noise inherent in the system, experimental measurement of such entangled states has proven difficult.

### Research

As decoherence due to noise is the main difficulty in performing experiments, a number of different cavity configurations is studied. Of particular interest are multiple cavities that are stacked on top of each other. Such cavity arrangement support more sophisticated pump schemes and thus are able to facilitate the separation of pump induced photoluminescence from the signal.