1. Data-driven Simulation of Urban Canopies
(jointly supervised PhD project of Heena Patel)
Urban areas (and areas with canopies in general) are our natural habitat and effects of human influence on this habitat are felt strongly here. This project aims at simulating tracer gas transport in such heterogeneous areas as well as on direct and data-driven multiscale flow simulations that potentially offer an alternative to DNS.
2. Adaptive Mesh Refinement in Single Components of Earth System Models
(jointly supervised PhD project of Yumeng Chen)
Developing new dynamical core for earth system models (ESM) is a very time consuming and complex task. In this project we investigate the option of integrating mesh adaptivity into single components of ESMs to improve accuracy. This enables the use of well-tested legacy code rather than developing a new adaptive dynamical core.
3. Stable Discretizations for Multiscale Systems
We also investigate the applicability to systems of multiscale PDEs. The problem here is that one usually has many different quantities that live in different function spaces like H(curl) or H(div). Coming up with stable pairings of finite elements for such problems is far from trivial. We combine ideas from differential geometry and multiscale finite elements to construct stable pairings by constructing a multiscale de Rham complex.
4. Semi-Lagrangian Multiscale Reconstruction
for Advection-Dominated Flows
We investigate generalizations of our idea to PDEs in a semi-Lagrangian setting in two and three dimensions. It turns out that semi-Lagrangian techniques can retain the “nice” part of the methods used in Project 1. One can reconstruct a basis using known information from previous time steps solving an inverse problem. This idea is scalable (strongly parallel), generalizable to higher dimensions, flexible in terms of elements and can work with real data. This will be the key technique to make it relevant to applications.
5. Flow-induced Coordinates for Multiscale
We use a simple 1-dimensional model for advection-dominated but yet diffusive tracer transport with multiple scales to demonstrate that correct boundary conditions for subgrid problems are essential and can be adjusted such that upscaling is meaningful in the sense that the way information travels is respected. The difficulty here is that subgrid problems need to be adjusted as well as their coupling to (coarse) computational scales. Our method is not practical but helps to understand, first, generalizabilty of our idea to higher dimensions and, secondly, it helps to come up with solutions for problems encountered using, for example, full Lagrangian techniques. Thirdly, our method suggests a way to separate subgrid problems strongly which makes our idea suitable for strong parallelization. The latter is clearly desired due to modern high-performance computer architectures.