Dissipative weak solutions for non-viscous single and multicomponent/phase flows

• Applicant: Privatdozent Dr. Philipp Öffner
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 520756621

The Euler equations of gas dynamics are used for many scientific and engineering problems - including the description of air flows - and involve the conservation of mass, momentum and total energy. For a long time, it was unclear whether multidimensional Euler equations are well-posed in the class of weak entropy solutions, i.e. whether there exists a unique solution that is continuously dependent on the initial data. This has now been disproven in several research papers and dissipative weak (DW) solutions were introduced as a generalization of the classic solution concept. They have established themselves as a meaningful solution concept. Some numerical methods have already been investigated with regard to their convergence properties for DW solutions. It was important that the numerical methods ensure the structural properties of Euler equations, e.g. positivity of density and pressure. This is the starting point of the project. First, a unified convergence analysis is performed. Especially, structure-preserving methods are examined and convergence properties with regard to DW solutions are demonstrated. For various finite element methods, convergence has only been proven in the semi-discrete case, i.e. the time was kept continuous. Inside the project, we further analyze convergence in the fully discrete case to close this open gap. In the second part of the project, we focus on non-viscous multicomponent/phase flows. They are used to describe complex models and include combinations of Euler equations with additional coupling terms. Already for convergence investigations regarding the Euler equations, the structure-preserving properties of the numerical procedure have been essential. In this part, novel structure-preserving schemes for non-viscous multicomponent/phase flows will be developed and analyzed in more detail. In the final part of the project, the concept of the DW solutions of the Euler equations will be extended to multicomponent/phase models, theoretically examined and a consistent solution theory will be provided. Subsequently, the convergence properties of the (novel) structure-preserving methods with regard to DW solutions are analyzed. The aim of the project is to give a uniform convergence analysis regarding DW solutions for the Euler equation, to construct novel structure-preserving methods and extend and establish the concept of DW in more general models.

• DFG Programme: Research Grants