Construction of weak solutions to compressible Navier--Stokes equations with general inflow/outflow boundary conditions via a numerical approximation

TL;DR

This paper develops a numerical scheme to construct weak solutions for the compressible Navier-Stokes equations with general inflow/outflow boundary conditions, filling a gap in the mathematical and computational understanding of such problems.

Contribution

It introduces a novel numerical approximation method for weak solutions under general boundary conditions, bridging the gap between theoretical existence proofs and computational approaches.

Findings

Proves convergence of the numerical scheme to weak solutions.

Extends the existence results to more general inflow/outflow boundary conditions.

Establishes a new link between numerical and theoretical analysis in fluid dynamics.

Abstract

The construction of weak solutions to compressible Navier-Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper [{\em Numer. Math.}, 125(3) : 441--510, 2013] for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient $\gamma>3$. Here we consider the same problem for the general non zero inflow-outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.