Abstract:
In this talk we introduce generalized 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 [1, 2, 3, 4]. In the case that the strong solution to the above equations exists, the dissipative solutions coincide with the strong solution on its life span [1]. 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 [5, 6]. The latter is the expected value of the dissipative measure-valued solutions and satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress. We will also discuss the question of error estimates and derive convergence rates for the Godunov finite volume method. The error analysis is realized by means of the relative energy which is a problem-suited “metric” [4]. Theoretical results will be illustrated by a series of numerical simulations.
This research has been done in cooperation with E. Feireisl (Prague), H. Mizerova (Bratislava), B. She (Beijing) and Y. Yuan (Nanjing). It was supported by the German Science Foundation under TRR/SFB 146 and by the Gutenberg Research College.
References
[1] E. Feireisl, M. Lukacova-Medvidova, H. Mizerova, B. She (2021): Numerical analysis of compressible fluid flows. Springer.
[2] E. Feireisl, M. Lukacova-Medvidova, H. Mizerova (2020): Convergence of finite volume schemes for the Euler equations via dissipative-measure valued solutions. Found. Comput. Math., 20, 923 – 966.
[3] E. Feireisl, M. Lukacova-Medvidova, H. Mizerova, B. She (2019): Convergence of a finite volume scheme for the compressible Navier-Stokes system. ESAIM: Math. Model. Num. 53, 1957 – 1979.
[4] M. Lukacova-Medvidova, B. She, Y. Yuan (2022): Error estimate of the Godunov method for multidimensional compressible Euler equations, J. Sci. Comput. 91, 71.
[5] E. Feireisl, M. Lukacova-Medvidova, H. Mizerova (2020): K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal. 40, 2227–2255.
[6] E. Feireisl, M. Lukacova-Medvidova, B. She, Y. Wang (2021): Computing oscillatory solutions of the Euler system via K-convergence, Math. Math. Models Methods Appl. Sci. 31, 537 – 576.
