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Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

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Abstract

This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (i.e., non-constant density and viscosity) incompressible Navier–Stokes system on a bounded domain. The proposed method is classical in the sense that it consists of a version of the Lax–Friedrichs explicit scheme for the transport equation and a version of Ladyzhenskaya’s implicit scheme for the Navier–Stokes equations. Under the condition that the initial density profile is strictly away from zero, the scheme is proven to be strongly convergent to a weak solution (up to a subsequence) within an arbitrary time interval, which can be seen as a proof of existence of a weak solution to the system. The results contain a new Aubin–Lions–Simon type compactness method with an interpolation inequality between strong norms of the velocity and a weak norm of the product of the density and velocity.

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Notes

  1. Throughout this paper, convergence of a numerical scheme to a weak solution of (1.1) means convergence up to a subsequence (due to an open problem on the uniqueness in three space-dimensions).

  2. The author is grateful to one of the reviewers for pointing out the reference.

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Acknowledgements

This paper was written during author’s one-year research stay in Fachbereich Mathematik, Technische Universität Darmstadt, Germany, with the grant Fukuzawa Fund (Keio Gijuku Fukuzawa Memorial Fund for the Advancement of Education and Research). The author expresses special thanks to Professor Dieter Bothe for his kind hosting in TU-Darmstadt. The author is supported by JSPS Grant-in-aid for Young Scientists #18K13443 and JSPS Grants-in-Aid for Scientific Research (C) #22K03391.

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  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Kohei Soga

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Soga, K. Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations. Numer. Math. (2024). https://doi.org/10.1007/s00211-024-01421-y

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