A finite difference method for inhomogeneous incompressible Navier-Stokes equations
Abstract
This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (non-constant density and viscosity) incompressible Navier-Stokes system on a bounded domain. The proposed method consists of a version of 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 , 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.
Cite
@article{arxiv.2302.14018,
title = {A finite difference method for inhomogeneous incompressible Navier-Stokes equations},
author = {Kohei Soga},
journal= {arXiv preprint arXiv:2302.14018},
year = {2023}
}