Seminar by Prof. Maria Lukacova-Medvidova, Uni Mainz, Germany on 19 March 2025

Speaker: Prof. Maria Lukacova-Medvidova, Institute of Mathematics, University of Mainz, Germany

Title: What do we approximate with the structure-preserving numerical methods for compressible flows?
Day, date and time: Wednesday, 19 March 2025, 16:00 hours
Venue: PSB1104

Abstract: In this talk, we introduce generalised solutions of compressible flows, the so-called dissipative solutions. We will concentrate on the inviscid flows, the Euler equations, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations. The dissipative solutions are obtained as a limit of suitable structure-preserving, consistent and stable finite volume schemes. In the case that the strong solution to the above equations exists, the dissipative weak solutions coincide with the strong solution on its life span. Otherwise, we apply a newly developed concept of K-convergence and prove the strong convergence of the empirical means of numerical solutions to a dissipative weak solution. The latter is the expected value of the dissipative measure-valued solutions and satises a weak formulation of the Euler equations modulo the Reynolds turbulent stress. If time permits, we will also derive error estimates for the corresponding finite volume method. The error analysis is realized using the relative energy, which is a problem-suited metric. Theoretical results will be illustrated by a series of numerical simulations.