Numerical Methods for Fluid Flow: High Order SBP Schemes, IMEX Advection-Diffusion Splitting and Positivity Preservation for Production-Destruction PDEs

The current demands regarding the numerical simulation of fluid flow often require highly accurate computations to obtain a detailed resolution of the occurring physical phenomena. The basic concept for the construction of a fluid solver is to transfer the physical model into a numerical scheme which complies with the underlying physical principles such as conservation and balances of certain quantities. In addition, the numerical methods are required to be stable and efficient. Again, stability is often determined by physically motivated quantities such as energy or entropy and it is generally easier to be achieved for low order schemes. Furthermore, the need for efficiency and possible implementation in parallel hardware environment has led to the development of sophisticated schemes in space with compact stencils such as discontinuous Galerkin (DG) methods and flux reconstruction schemes which extend classical space discretization methods. In this work, the newly found generalized SBP properties of DG schemes on Legendre-Gauss nodes pave the way to their application to skew-symmetric forms. Since their quadrature rule possesses a higher degree of exactness, DG schemes on Legendre-Gauss nodes are usually more accurate than those on Legendre-Gauss-Lobatto nodes and might be preferable for long-time simulations, which is precisely the situation in which the preservation of secondary quantities should be most beneficial.