Error estimates for the implicit MAC scheme for the compressible Navier–Stokes equations

We prove existence of a solution to the implicit MAC scheme for the compressible Navier–Stokes equations. We derive error estimates for this scheme on two and three dimensional Cartesian grids. Error estimates are obtained by using the discrete version of the relative energy method introduced on the continuous level in Feireisl et al. (J Math Fluid Mech 14(4):717–730, 2012). A systematic use of the theoretical “continuous” analysis of the equations in combination with the numerical tools is crucial for the result. This error estimate does not uses stability hypotheses on the solution of the numerical scheme.

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