Institute of Mathematics


Modul:   MAT870  Zurich Colloquium in Applied and Computational Mathematics

How to compute oscillatory solutions of the Euler equations

Prof. Dr. Maria Lukacova talk

Speaker invited by: Prof. Dr. Rémi Abgrall

Date: 26.02.20  Time: 16.15 - 17.45  Room: Y27H46

An iconic example of hyperbolic conservation laws are the Euler equations of gas dynamics expressing the conservation of mass, momentum and energy. Recently, it has been shown that even for smooth initial data the Euler equations may have infinitely many physically admissible, i.e. weak entropy, solutions. Related to the ill-​posedness of the Euler equations is the observation that approximate solutions obtained by standard finite volume methods may develop oscillations and cascades of new small scale substructures. Consequently, a fundamental question is: What is the limit of numerical solutions as mesh parameter is refined? 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 the numerical solutions obtained 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 its observable quantities, such as the mean and variance and proof their strong convergence in space and time. Theoretical results will be 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.