Implicit MAC scheme for compressible Navier-Stokes equations: Unconditional error estimates

Thierry Gallouët (1) , Raphaele Herbin (1) , David Maltese (2) , Antonin Novotny (2)

1 I2M - Institut de Mathématiques de Marseille
2 IMATH - Institut de Mathématiques de Toulon - EA 2134

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. These estimates are unconditional: They are uniform with respect to the space and time discretization with coefficient independent on the numerical solution. Error estimates are obtained by using the discrete version of the relative energy method introduced on the continuous level in [16]. A systematic use of the theoretical "continuous" analysis of the equations in combination with the numerical tools is crucial for the result. This is the first proof ever of the inconditional error estimate of the MAC scheme for the compressible Navier-Stokes equations.

Mots clés: Compressible fluids; Navier-Stokes equations; Cartesian grids; Marker and Cell scheme; Error estimates; AMS classification 35Q30; 65N12; 76N10; 76N15; 76M12; 76M20

Domaines: Analyse numérique [math.NA]

Thierry Gallouët, Raphaele Herbin, David Maltese, Antonin Novotny. Implicit MAC scheme for compressible Navier-Stokes equations: Unconditional error estimates. 2017. ⟨hal-01448698⟩