1. Numerical simulation of quantum systems
The research focus of the Arnold group is on the mathematical analysis and the development of efficient numerical methods for quantum mechanical simulations with applications to quantum wave guides and quantum transport through tunneling structures.
Crucial numerical aspects that are being tackled by this research group include transparent boundary conditions (for the Schr¨odinger equation, e.g.) and WKBdiscretizations for highly oscillatory solutions. The formulation and stable discretization of open or transparent boundary conditions are important for accurate simulations of quantum scattering problems and for reducing the simulation domain in order to increase the computational efficiency. Such boundary conditions are non-local in time and in space (in 2 or 3 dimensional situations). Hence, their numerical treatment is very delicate and can be time consuming. The group around Arnold has developed novel approaches for such boundary conditions in conjunction with finite difference schemes (for planar, wave guide and circular geometries).
Highly oscillatory differential equations appear in many wave propagation problems, like quantum mechanical electron transport, e.g. Standard discretizations schemes (be it finite differences or FEMs) have to resolve (already for 1D problems) each oscillation with at least 10 grid points which can lead to substantial numerical costs for non-linearly coupled systems. The goal of WKB-based numerical schemes is to develop efficient and, simultaneously, highly accurate numerical schemes, which do not need to resolve each oscillation. This can be achieved by eliminating the dominant oscillation frequency, if the resulting oscillatory integrals are discretized appropriately. The key aspect is here to resolve the weak limit of the Schroedinger solution (i.e. the semi-classical limit h --> 0) correctly, even for a constant grid spacing! Such schemes are currently being developed, and they appear highly supperior to existing techniques (both with respect to accuracy and computational effort).
The research of Othmar Koch focusses on efficient numerical methods for the multi-configuration time-dependent Hartree (MCTDH) and Hartree-Fock (MCTDHF) methods with applications in photonics (ultrafast laser pulses) and in quantum chemistry (quantum molecular dynamics). Of particular importance are here geometric numerical integration schemes that preserve, on the discrete level, some of the conserved quantities of the continuous system. Moreover, the approximation of the meanfield terms appearing in the MCTDH(F) equations by low rank operators can be improved by using hierarchical matrices, since this constitutes the computationally most demanding part of the numerical solution methods.
Fruitful cooperations between Arnold and the group of J¨ungel have been existing for many years and are funded by the FWF via the Wissenschaftskolleg “Differential Equations”, for example. Within the CompMat consortium new cooperations are planed with the groups of Held and Burgd¨orfer (numerical simulations of quantum systems, in particular low dimensional structures). Moreover, Dr. Koch’s research on numerical aspects of MCTDHF models (in particular approximation strategies for nonlocal (integral) operators) is of high interest in many physics and chemistry applications. Within the CompMat consortium it lends itself to cooperations with the groups of Melenk and Blaha.