Positivity preserving methods for advection-dominated problems in the framework of partial differential-algebraic equations

The mathematical description of problems in Computational Fluid Dynamics typically leads to hyperbolic partial differential equations (PDEs). Physical factors as well as the numerical simulation process can cause additionally constraints on the PDE. Typical examples are fluid-structure interaction problems, magnetohydrodynamic equations, the finite element discretization of the incompressible Navier–Stokes equations or the usage of a computational domain including hanging nodes in adaptive finite element simulations. But, much simpler problems belong to this problem class as well. The inclusion of inflow boundary conditions in a strong way by Lagrange multipliers in advection-dominated PDEs leads to partial differential-algebraic equation. Therefore, we consider the class of partial differential-algebraic equations (PDAEs) with an advection-dominated part. Numerical simulation of advection-dominated partial differential-algebraic equations is still a great challenge in numerical analysis research. Unphysical oscillations can occur in the numerical solution of advection-dominated PDEs, e. g., due to non-smooth initial data. On the other hand the numerical treatment of differential-algebraic equations differs strongly from the treatment of ordinary differential equations. Both topics have been in focus of intensive research in the last decades and new numerical algorithms capable to handle the occuring problems were developed.