How to compute oscillatory solutions of the compressible Euler equations?
A talk in the Oberseminar Numerik series by
Maria Lukacova
| Abstract: | It is a well-known fact that the Euler equations of
gas dynamics are ill-posed in the class of
physically admissible, i.e. weak entropy, solutions.
Consequently even for smooth initial data we
may have infinitely many solutions. In particular,
approximate solutions obtained by suitable numerical
schemes may develop oscillations that are visualized
as a cascade of small scale structures. In the present talk we will present a concept of K-convergence that can be seen as a new tool in numerical analysis of the ill-posed problems, such as the Euler equations. We will show that numerical solutions generated by some standard finite volume methods generate a dissipative measure-valued solution, which is an appropriate probability measure (Young measure). We will also show how to effectively compute the observable quantities, such as the mean and variance and proof rigorously the strong convergence in space and time and in the Wasserstein distance. Theoretical results will be also illustrated by a series of numerical simulations. The present research has been done in collaboration with E. Feireisl (Prague/Berlin), H. Mizerova (Bratislava), B. She (Prague) and Y. Wang (Beijing). It has been partially supported by TRR 146 Multiscale simulation methods for soft matter systems and by TRR 165 Waves to weather funded by DFG. |