Technische Wissenschaften, Physik, Mathematik
Geometric methods of discretization and parameterization
Applied mathematics
University of British Columbia, McGill University, Memorial University of Newfoundland, University of Vienna
01.05.2014 - 30.04.2017

This project is devoted to the study of discretization and parameterization schemes preserving symmetries and conservation laws. It will focus both on the theoretical foundations of the techniques to be used as well as on a comprehensive numerical testing of the developed schemes.

Several methods, such as the discretization in computational coordinates, the invariantization using equivariant moving frames and evolution-projection techniques will be applied to construct mesh-based and meshless invariant discretizations. Techniques from the field of mimetic discretization will be enhanced to also allow preserving certain conservation laws in our invariant numerical schemes.

The proposed methods and algorithms for finding invariant discretizations will be tested with several important physical models, including the one- and two-dimensional wave equations, elliptic Monge-Ampere equations as well as the two-dimensional Euler equations. Conservative invariant discretizations will be constructed for the two-dimensional Euler equations and the shallow-water equations, for which we have already constructed invariant discretization schemes.

Within the problem of invariant and conservative parameterization, we will focus on the problem of geometry-preserving turbulence modeling for both two- and three-dimensional incompressible fluids. New invariant and conservative hyperdiffusion models will be constructed and tested for the two-dimensional Euler equations. Invariantizations of existing classical subgrid-scale closure models for the three-dimensional filtered Navier-Stokes will be carried out. The resulting invariant closure models will be tested numerically by implementing them within the OpenFoam CFD software package.