Abstract
We review our recent results on the convergence of invariant domain-preserving finite volume solutions to the Euler equations of gas dynamics. If the classical solution exists we obtain strong convergence of numerical solutions to the classical one applying the weak-strong uniqueness principle. On the other hand, if the classical solution does not exist we adapt the well-known Prokhorov compactness theorem to space-time probability measures that are generated by the sequences of finite volume solutions and show how to obtain the strong convergence in space and time of observable quantities. This can be achieved even in the case of ill-posed Euler equations having possibly many oscillatory solutions.
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Notes
- 1.
Here the mean value \(\left\langle \mathscr {V}_{t,x}; b\left( \tilde{\varvec{U}}\right) \right\rangle \equiv \int _{\mathbb {R}^{d+2}} b\left( \tilde{\varvec{U}} \right) d \mathscr {V}_{t,x}( \tilde{\varvec{U}} )\) for \(\varvec{U} \in \mathbb {R}^{d+2}\) and b bounded continuous function.
- 2.
We recall that the Wasserstein metric of q-th order, \(q \in [1, \infty )\), is defined in the following way \(W_q({\mathscr {N}}, {\mathscr {V}}):= \left\{ \inf _{\pi \in \Pi (\mathscr {N}, \mathscr {V})} \int _{\mathbb {R}^{d+2} \times \mathbb {R}^{d+2}} | \zeta -\xi |^q d\pi (\zeta ,\xi ) \right\} ^{1/q}\), where \(\Pi (\mathscr {N}, \mathscr {V})\) is the set of probability measures on \(\mathbb {R}^{d+2} \times \mathbb {R}^{d+2}\) with marginals \(\mathscr {N}\) and \(\mathscr {V}\).
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Acknowledgements
The author wishes to thank E. Feireisl (Prague), H. Mizerová (Bratislava) and B. She (Prague) for fruitful discussions on the topic. This research was supported by the German Science Foundation under the grants TRR 146 Multiscale simulation methods for soft matter systems and TRR 165 Waves to Weather.
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Lukáčová-Medvid’ová, M. (2020). \(\mathscr {K}\)-Convergence of Finite Volume Solutions of the Euler Equations. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_2
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