I am working on numerical methods for compressible and incompressible flows. I focus on Finite Difference (FD) and Finite-Element Based Schemes (FE) like Continuous and Discontinous Galerkin (CG/DG), Flux Reconstruction (FR) and Residual Distribution (RD). I analyze the stability and approximation properties of these schemes. One major tool
are summation-by-parts operators to transfrom results from the continuous setting to the discrete framework. Further, I also develop new stable methods for advection-dominated problems.
Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability without a time integration method is studied. In my reserach, I focus on this task and construct fully discrete entropy stable schemes. By applying the relaxation technique proposed by D. Ketcheson or the application of modal filters/artificial viscosity, I am able to construct fully discrete, explicit entropy stable schemes. Besides this, I am investigating stability properties of DeC, Ader and RK methods and focus on the relations between these schemes.
As another point, I am working on positivity preserving time integration methods and was able to develop an
arbitrary high-order, conservative and positivity preserving modified Patankar DeC scheme. In the future, I will investigate the stability properties of these modified Patankar schemes.
The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy) weak solutions. Therefore, the concept of measure valued solutions seems nowadays more suitable for the analysis. In Mainz, I study the convergence of entropy stable FE based schemes for the complete compressible Euler equations in the multidimensional case and try to show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Therefore, I have to work on stability and consistency estimates in this context.