Convergence and error estimates for a finite difference scheme for the multi-dimensional compressible Navier-Stokes system

Abstract: We prove convergence of a finite difference approximation of the compressible Navier--Stokes system towards the strong solution in $R^d,$ $d=2,3,$ for the adiabatic coefficient $\gamma>1$. Employing the relative energy functional, we find a convergence rate which is \emph{uniform} in terms of the discretization parameters for $\gamma \geq d/2$. All results are \emph{unconditional} in the sense that we have no assumptions on the regularity nor boundedness of the numerical solution. We also provide numerical experiments to validate the theoretical convergence rate. To the best of our knowledge this work contains the first unconditional result on the convergence of a finite difference scheme for the unsteady compressible Navier--Stokes system in multiple dimensions.