Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations

Herbin, Raphaèle; Latché, Jean-Claude; Nguyen, Trung Tan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 893-944.

In this paper, we build and analyze the stability and consistency of decoupled schemes, involving only explicit steps, for the isentropic Euler equations and for the full Euler equations. These schemes are based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered. The velocity convection term is built in such a way that the solutions satisfy a discrete kinetic energy balance, with a remainder term at the left-hand side which is shown to be non-negative under a CFL condition. In the case of the full Euler equations, we solve the internal energy balance, to avoid the space discretization of the total energy, whose expression involves cell-centered and face-centered variables. However, since the residual terms in the kinetic energy balance (probably) do not tend to zero with the time and space steps when computing shock solutions, we compensate them by corrective terms in the internal energy equation, to make the scheme consistent with the conservative form of the continuous problem. We then show, in one space dimension, that, if the scheme converges, the limit is indeed an entropy weak solution of the system. In any case, the discretization preserves by construction the convex of admissible states (positivity of the density and, for Euler equations, of the internal energy), under a CFL condition. Finally, we present numerical results which confort this theory.

Classification : 35Q31, 65N12, 65M12, 76M10, 76M12
Mots clés : Finite volumes, staggered grid, Euler equations, isentropic barotropic, compressible flows, shallo water, analysis.

[1] G. Ansanay-Alex, F. Babik, J.-C. Latché and D. Vola, An L²-stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements. Int. J. Numer. Meth. Fluids 66 (2011) 555–580. | DOI | MR | Zbl

[2] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws. Birkhauser, Basel (2004). | DOI | MR | Zbl

[3] CALIF³S, A software components library for the computation of reactive turbulent flows. Available at: https://gforge.irsn.fr/gf/project/isis (2018).

[4] P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II. Edited by P. Ciarlet and J.L. Lions. North Holland, Amsterdam, Oxford (1991) 17–351. | MR | Zbl

[5] M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Série Rouge 7 (1973) 33–75. | Numdam | MR | Zbl

[6] L. Gastaldo, R. Herbin, W. Kheriji, C. Lapuerta and J.-C. Latché, Staggered discretizations, pressure correction schemes and all speed barotropic flows, in Finite Volumes for Complex Applications VI – Problems and Perspectives – Prague, Czech Republic, Vol. 2 (2011) 39–56. | MR | Zbl

[7] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, Berlin Heidelberg (1996). | DOI | MR | Zbl

[8] D. Grapsas, R. Herbin, W. Kheriji and J.-C. Latché, An unconditionally stable staggered pressure correction scheme for the compressible Navier–Stokes equations. SMAI – J. Comput. Math. 2 (2016) 51–97. | DOI | MR | Zbl

[9] J.L Guermond and R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. Comptes Rendus de l’Académie des Sciences de Paris – Série I – Analyse Numérique 346 (2008) 801–806. | MR | Zbl

[10] J.L Guermond, R. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230 (2011) 4248–4267. | DOI | MR | Zbl

[11] F.H. Harlow and A.A. Amsden, A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys. 8 (1971) 197–213. | DOI | Zbl

[12] F.H. Harlow and J.E. Welsh, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8 (1965) 2182–2189. | DOI | MR | Zbl

[13] R. Herbin and J.-C. Latché, Kinetic energy control in the MAC discretization of the compressible Navier–Stokes equations. Int. J. Finite Vol. 7 (2010) 41. | MR | Zbl

[14] R. Herbin, W. Kheriji and J.-C. Latché, Staggered schemes for all speed flows, in Congrès National de Mathématiques Appliquées et Industrielles. ESAIM: Proc. 35 (2011) 122–150. | MR | Zbl

[15] R. Herbin, J.-C. Latché and T.T. Nguyen, Explicit staggered schemes for the compressible Euler equations. ESAIM: Proc. 40 (2013) 83–102. | DOI | MR | Zbl

[16] R. Herbin, W. Kheriji and J.C. Latché, On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations. ESAIM: M2AN 48 (2014) 1807–1857. | DOI | MR | Zbl

[17] R. Herbin, J.-C. Latché and N. Therme, On a class of consistent staggered schemes for the compressible Euler equations (2016).

[18] B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95 (1991) 59–84. | DOI | MR | Zbl

[19] M.-S. Liou, A sequel to AUSM, part II: AUSM+-up. J. Comput. Phys. 214 (2006) 137–170. | DOI | MR | Zbl

[20] M.-S. Liou and C.J. Steffen, A new flux splitting scheme. J. Comput. Phys. 107 (1993) 23–39. | DOI | MR | Zbl

[21] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. Diff. Eq. 8 (1992) 97–111. | DOI | MR | Zbl

[22] N. Therme, Schémas numériques pour la simulation de l’explosion. Ph.D. Thesis, Aix Marseille Univ., Marseille (2015).

[23] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics – A Practical Introduction, 3rd edn. Springer, Berlin Heidelberg (2009). | DOI | MR | Zbl