Dellacherie, Stéphane Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. (English) Zbl 1329.76228 J. Comput. Phys. 229, No. 4, 978-1016 (2010). Summary: We propose a theoretical framework to clearly explain the inaccuracy of Godunov type schemes applied to the compressible Euler system at low Mach number on a Cartesian mesh. In particular, we clearly explain why this inaccuracy problem concerns the 2D or 3D geometry and does not concern the 1D geometry. The theoretical arguments are based on the Hodge decomposition, on the fact that an appropriate well-prepared subspace is invariant for the linear wave equation and on the notion of first-order modified equation. This theoretical approach allows to propose a simple modification that can be applied to any collocated scheme of Godunov type or not in order to define a large class of collocated schemes accurate at low Mach number on any mesh. It also allows to justify collocated schemes that are accurate at low Mach number as, for example, the Roe-Turkel and the AUSM\(^{+}\)-up schemes, and to find a link with a collocated incompressible scheme stabilized with a Brezzi-Pitkäranta type stabilization. Numerical results justify the theoretical arguments proposed in this paper. Cited in 2 ReviewsCited in 98 Documents MSC: 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids Keywords:compressible Euler system; low Mach number flow; Roe scheme; Roe-Turkel scheme; AUSM\(^{+}\)-up scheme; linear wave equation; Hodge decomposition Software:AUSM PDFBibTeX XMLCite \textit{S. Dellacherie}, J. Comput. Phys. 229, No. 4, 978--1016 (2010; Zbl 1329.76228) Full Text: DOI References: [1] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comp. Phys., 43, 357-372 (1981) · Zbl 0474.65066 [2] Massella, J.-M.; Faille, I.; Gallouët, T., On an approximate Godunov scheme, Int. J. Comput. Fluid. Dyn., 12, 133-149 (1999) · Zbl 0944.76041 [3] Buffard, T.; Gallouët, T.; Hérard, J.-M., A sequel to a rough Godunov scheme: application to real gases, Comput. Fluids, 29, 813-847 (2000) · Zbl 0961.76048 [4] F. Beux, M.V. Salvetti, E. Sinibaldi, A preconditioned implicit Roe’s scheme for barotropic flows: towards simulation of cavitation phenomena, Technical Report 4891, INRIA, 2003.; F. Beux, M.V. Salvetti, E. Sinibaldi, A preconditioned implicit Roe’s scheme for barotropic flows: towards simulation of cavitation phenomena, Technical Report 4891, INRIA, 2003. [5] Clerc, S., Numerical simulation of the homogeneous equilibrium model for two-phase flow, J. Comp. Phys., 161, 354-375 (2000) · Zbl 0965.76051 [6] F. Dauvergne, J.-M. Ghidaglia, F. Pascal, J.-M. Rovarch, Renormalization of the numerical diffusion for an upwind finite volume method. Application to the simulation of Kelvin-Helmholtz instability, in: R. Eymard, J.-M. Hérard (Eds.), Proceedings of the Fifth International Symposium on Finite Volumes for Complex Applications, Wiley, 2008, pp. 321-328.; F. Dauvergne, J.-M. Ghidaglia, F. Pascal, J.-M. Rovarch, Renormalization of the numerical diffusion for an upwind finite volume method. Application to the simulation of Kelvin-Helmholtz instability, in: R. Eymard, J.-M. Hérard (Eds.), Proceedings of the Fifth International Symposium on Finite Volumes for Complex Applications, Wiley, 2008, pp. 321-328. · Zbl 1374.76128 [7] Guillard, H.; Viozat, C., On the behavior of upwind schemes in the low Mach number limit, Comput. Fluids, 28, 63-86 (1999) · Zbl 0963.76062 [8] Guillard, H.; Murrone, A., On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes, Comput. Fluids, 33, 655-675 (2004) · Zbl 1049.76040 [9] Guillard, H.; Murrone, A., Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model, Comput. Fluids, 37, 10, 1209-1224 (2008) · Zbl 1237.76089 [10] Paillère, H.; Viozat, C.; Kumbaro, A.; Toumi, I., Comparison of low Mach number models for natural convection problems, Heat and Mass Transfer, 36, 567-573 (2000) [11] Volpe, G., Performance of compressible flow codes at low Mach number, AIAA J., 31, 1, 49-56 (1993) · Zbl 0775.76140 [12] Liou, M.-S., A sequel to AUSM: \(AUSM^+\), Part II: \(AUSM^+\)-up for all speeds, J. Comp. Phys., 214, 1, 137-170 (2006) · Zbl 1137.76344 [13] Sabanca, M.; Brenner, G.; Alemdarogˇlu, N., Improvements to compressible Euler methods for low-Mach number flows, Int. J. Numer. Meth. Fluids, 34, 167-185 (2000) · Zbl 0995.76056 [14] Turkel, E., Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comp. Phys., 72, 277-298 (1987) · Zbl 0633.76069 [15] Turkel, E., Review of preconditioning methods for fluid dynamics, Appl. Numer. Math., 12, 257-284 (1993) · Zbl 0770.76048 [16] Mary, I.; Sagaut, P., Large Eddy simulation of flow around an airfoil near stall, AIAA J., 40, 6, 1139-1145 (2002) [17] Li, X.-S.; Gu, C.-W., An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour, J. Comp. Phys., 227, 5144-5159 (2008) · Zbl 1388.76207 [18] Li, X.-S.; Gu, C.-W.; Xu, J.-Z., Development of Roe-type scheme for all-speed flows based on preconditioning method, Comput. Fluids, 38, 810-817 (2009) · Zbl 1242.76171 [19] Schochet, S., Fast singular limits of hyperbolic PDEs, J. Differ. Eqs., 114, 476-512 (1994) · Zbl 0838.35071 [20] Perthame, B., Boltzmann type schemes for the gas dynamics and the entropy property, SIAM J. Num. Anal., 27, 1405-1421 (1990) · Zbl 0714.76078 [21] Eymard, R.; Herbin, R.; Latché, J.-C., On a stabilized colocated finite volume scheme for the Stokes problem, Math. Model. Numer. Anal., 40, 3, 501-527 (2006) · Zbl 1160.76370 [22] A. Majda, in: Springer-Verlag (Ed.), Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences Series, Wiley, New York, 1984, pp. 30-80.; A. Majda, in: Springer-Verlag (Ed.), Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences Series, Wiley, New York, 1984, pp. 30-80. · Zbl 0537.76001 [23] F. Boyer, P. Fabrie, in: Éléments d’analyse pour l’Étude de quelques modèles d’Écoulements de fluides visqueux incompressibles, Mathématiques & Applications, Springer, French, 2006 (Chapter 2).; F. Boyer, P. Fabrie, in: Éléments d’analyse pour l’Étude de quelques modèles d’Écoulements de fluides visqueux incompressibles, Mathématiques & Applications, Springer, French, 2006 (Chapter 2). · Zbl 1105.76003 [24] Minion, M. L., A projection method for locally refined grids, J. Comp. Phys., 127, 158-178 (1996) · Zbl 0859.76047 [25] Godlewski, E.; Raviart, P.-A., (Mardsen, J.; Sirovich, L.; John, F., Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 (1996), Springer-Verlag: Springer-Verlag New York), 215-220 [26] Thornber, B. J.R.; Drikakis, D., Numerical dissipation of upwind schemes in low Mach flow, Int. J. Numer. Meth. Fluids, 56, 1535-1541 (2008) · Zbl 1136.76034 [27] R.D. Richtmyer, K.W. Morton, in: Krieger (Ed.), Difference Methods for Initial-Value Problems, second ed., 1994.; R.D. Richtmyer, K.W. Morton, in: Krieger (Ed.), Difference Methods for Initial-Value Problems, second ed., 1994. · Zbl 0824.65084 [28] S. Dellacherie, Checkerboard modes and wave equation, in: Proceedings of the 18th Conference on Scientific Computing, Podbanske, Slovakia, 2009, pp. 71-80.; S. Dellacherie, Checkerboard modes and wave equation, in: Proceedings of the 18th Conference on Scientific Computing, Podbanske, Slovakia, 2009, pp. 71-80. · Zbl 1173.76027 [29] Beccantini, A., Colella-Glaz splitting scheme for thermally perfect gases, (Toro, E. F., Godunov Methods: Theory and Application (2001), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers New York), 89-95 · Zbl 1064.76553 [30] Colella, P.; Glaz, H., Efficient solution algorithms for the Riemann problem for real gases, J. Comp. Phys., 59, 264-289 (1985) · Zbl 0581.76079 [31] Birken, P.; Meister, A., Stability of preconditioned finite volume schemes at low Mach numbers, BIT Numer. Math., 45, 3, 463-480 (2005) · Zbl 1124.76038 [32] Liou, M.-S., A sequel to AUSM: \(AUSM^+\), J. Comp. Phys., 129, 2, 364-382 (1996) · Zbl 0870.76049 [33] Liou, M.-S.; Steffen, C. J., A new flux splitting scheme, J. Comp. Phys., 107, 1, 23-39 (1993) · Zbl 0779.76056 [34] F. Brezzi, J. Pitkäranta, On the Stabilization of Finite Element Approximation of the Stokes Equation, in: W. Hackbusch (Ed.), Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, vol. 10, 1984, pp.11-19.; F. Brezzi, J. Pitkäranta, On the Stabilization of Finite Element Approximation of the Stokes Equation, in: W. Hackbusch (Ed.), Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, vol. 10, 1984, pp.11-19. [35] S. Dellacherie, P. Omnes, F. Rieper, The influence of cell geometry on the Godunov scheme applied to the linear wave equation, submitted for publication.; S. Dellacherie, P. Omnes, F. Rieper, The influence of cell geometry on the Godunov scheme applied to the linear wave equation, submitted for publication. · Zbl 1206.65208 [36] Rieper, F., Influence of cell geometry on the behaviour of the first-order roe scheme in the low Mach number regime, (Eymard, R.; Hérard, J.-M., Finite Volumes for Complex Applications V (2008), Wiley), 625-632 · Zbl 1374.76169 [37] Rieper, F.; Bader, G., The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime, J. Comp. Phys., 228, 8, 2918-2933 (2009) · Zbl 1159.76027 [38] Vidović, D.; Segal, A.; Wesseling, P., A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids, J. Comp. Phys., 217, 277-294 (2006) · Zbl 1101.76037 [39] Davis, G. D.V., Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids, 3, 249-264 (1983) · Zbl 0538.76075 [40] P.L.Q. et al., Modeling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 1. Reference Solutions, Math. Model. Numer. Anal. 39 (3) (2005a) 609-616.; P.L.Q. et al., Modeling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 1. Reference Solutions, Math. Model. Numer. Anal. 39 (3) (2005a) 609-616. · Zbl 1130.76047 [41] H.P. et al., Modeling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 2. Contributions to the June 2004 Conference, Math. Model. Numer. Anal. 39 (3) (2005b) 617-621.; H.P. et al., Modeling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 2. Contributions to the June 2004 Conference, Math. Model. Numer. Anal. 39 (3) (2005b) 617-621. 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