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We present new error estimates for the finite volume and finite differencemethods applied to the compressible Navier-Stokes equations. The maininnovative ingredients of the improved error estimates are a refinedconsistency analysis combined with a continuous version of the relative energyinequality. Consequently, we obtain better convergence rates than thoseavailable in the literature so far. Moreover, the error estimates hold in thewhole physically relevant range of the adiabatic coefficient.

Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system