Prof. Dr. Gregor Gassner, Colloquium, Trends in Scientific Computing, TU Dortmund, Dortmund, Germany:
For marginally resolved simulations of non-linear PDEs, such as the compressible Navier-Stokes (NS) equations or the magnetohydrodynamics (MHD) equations, we observe that high order methods are prone to aliasing instabilities, which can lead to total failure and breakdown of the algorithms. Aliasing issues are provoked by insufficient discrete integration precision, collocation of non-linear terms, polynomial ansatz spaces for rational functions, and, again, by insufficient grid resolution.
The aim of this talk is to discuss a remedy for aliasing issues and present its connection to discrete product and chain rules. By using building blocks from summation-by-parts finite difference, first order finite volume and Gauss-Lobatto-Legendre spectral element methods, we show how to construct nodal high order discontinuous Galerkin methods that are discretely kinetic energy preserving and/or entropy stable on unstructured hexahedral curvilinear grids in three spatial dimensions for the compressible NS equations, the resistive MHD equations, and similar nonlinear systems.
