Fast numerical methods for large-scale multiphysics problems

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The first type of problems to be considered in this project are linear and nonlinear fluid-structure interaction problems. They appear in a whole range of applications starting from aircraft design, to simulation of flows in human arteries, heart, and the alveoli of the airways. The intention is to explore a new approach to such problems that, to our knowledge, has not been used so far. The key idea of this approach is to reformulate both, the fluid and the structure problems in terms of a new variable, the stress tensor. Further, by using a proper splitting approach, the 3D time-discrete system of equations can be reduced to a set of much simpler, lower-dimensional problems that can be resolved much easier. So, the resulting numerical algorithm will be significantly faster and easier to implement on parallel computers. Another very pleasant outcome of this new setting of fluid-structure interaction problems is that since in both domains it uses only the stress as a variable, it does not require explicit tracking of the shape of the structure domain that is usually quite problematic in the existing methods in primitive variables. The second major problem in this project is about the construction of a fast numerical method for solving the equations describing the dynamics of the atmosphere and the oceans in a spherical shell. The obvious application is to study the flow in the Earth's atmosphere and oceans in a coupled fashion. Since the size of the problem is enormous, the algorithm to be designed must be usable on very large parallel clusters of computers. I plan to use a high order incompressible method devised recently in my group and modify it to relax the incompressibility constraint in the upper atmosphere. It has been verified to be very optimal, particularly on large parallel computers. So, the algorithm to be developed will be very fast and will allow for simulations with a very high resolution. The biggest challenges of its design are twofold. On one hand, it will require to use meshes with local refinement which is not easy to combine with direction splitting algorithms that we also intend to employ. Recently, in my group we developed an approach for the design of such algorithms using hanging nodes and proved that it is unconditionally stable in case of parabolic problems. This approach will be further extended to include an efficient parallelization strategy (for the use on parallel clusters). The other challenge comes from the geometry of the domain, a very thin shell, that poses particular difficulties to parallelization. This will require the use of graph-partitioning methods. The resulting methods and software from this part of the project will be qualitatively superior to most of the currently available numerical models that are hydrostatic i.e. they incorporate the dynamics of the flow in the vertical direction in a very simplified manner. Besides, they will allow for much more refined simulations.