Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation

Many numerical methods for fluid dynamics are suitable only for a single, idealized type of fluid. Most prominently, algorithms for compressible flow are often tailored to ideal gases and another class of schemes is designed for incompressible media. This dissertation targets a numerical method for the simulation of the flow of fluids with differing thermodynamical properties, in particular both compressible and incompressible fluids. By considering the Navier--Stokes equations in terms of certain generalized variable sets, the singularity of the incompressible limit, which arises when using conservation variables, can be avoided. Especially entropy variables, which stem from symmetrization theory, can ensure desirable properties like the fulfillment of the second law of thermodynamics. To obtain a discretization that allows local adaptation and high geometric flexibility, a space-time discontinuous Galerkin finite element method is developed. Particular attention is paid to the examination which components of the numerical method have to be adapted when using specific thermodynamical models. Numerical results for a diverse range of compressible and incompressible test cases underline the applicability of the method for various fluids and conditions. Apart from the numerical algorithms, the project also involved the further development of the object-oriented general-purpose discontinuous Galerkin finite element software framework hpGEM. This thesis presents aspects of its philosophy and design, and discusses examples of how the components can be applied. The formulation in terms of general sets of variables, a mathematical method that accommodates geometric flexibility and adaptivity, and a software environment for the implementation of numerical methods are important parts for the construction of finite element methods for gas-liquid multiphase flows