On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, p. 1807-1857
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher-Turek or Crouzeix-Raviart finite elements. We first show that a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropic case, a solution also satisfies a discrete elastic potential balance; integrating these equations over the domain readily yields discrete counterparts of the stability estimates which are known for the continuous problem. In the case of the full Euler equations, the scheme relies on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with the original total energy conservative equation is obtained thanks to corrective terms in the internal energy balance, designed to compensate numerical dissipation terms appearing in the discrete kinetic energy inequality. It is then shown in the 1D case, that, supposing the convergence of a sequence of solutions, the limit is an entropy weak solution of the continuous problem in the barotropic case, and a weak solution in the full Euler case. Finally, we present numerical results which confirm this theory.

DOI : https://doi.org/10.1051/m2an/2014021
Classification:  35Q31,  65N12,  76M10,  76M12
Keywords: finite volumes, finite elements, staggered, pressure correction, Euler equations, shallow-water equations, compressible flows, analysis
@article{M2AN_2014__48_6_1807_0,
author = {Herbin, R. and Kheriji, W. and Latch\'e, J.-C.},
title = {On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {6},
year = {2014},
pages = {1807-1857},
doi = {10.1051/m2an/2014021},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_6_1807_0}
}

Herbin, R.; Kheriji, W.; Latché, J.-C. On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1807-1857. doi : 10.1051/m2an/2014021. http://www.numdam.org/item/M2AN_2014__48_6_1807_0/

[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. Methods Fluids 66 (2011) 555-580. | MR 2839213 | Zbl pre05919101

[2] F. Archambeau, J.-M. Hérard and J. Laviéville, Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases. Comput. Fluids 38 (2009) 1495-1509. | MR 2645756 | Zbl 1242.76152

[3] R. Berry, Notes on PCICE method: simplification, generalization and compressibility properties. J. Comput. Phys. 215 (2006) 6-11. | MR 2215649 | Zbl 1140.76412

[4] H. Bijl and P. Wesseling, A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141 (1998) 153-173. | MR 1619651 | Zbl 0918.76054

[5] CALIF3S. A software components library for the computation of reactive turbulent flows. Available on https://gforge.irsn.fr/gf/project/isis.

[6] V. Casulli and D. Greenspan, Pressure method for the numerical solution of transient, compressible fluid flows. Int. J. Numer. Methods Fluids 4 (1984) 1001-1012. | Zbl 0549.76050

[7] A. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | MR 242392 | Zbl 0198.50103

[8] P.G. Ciarlet, Basic error estimates for elliptic problems, in vol. II of Handb. Numer. Anal. Edited by P. Ciarlet and J. Lions. North Holland (1991) 17-351. | MR 1115237 | Zbl 0875.65086

[9] M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Série Rouge 7 (1973) 33-75. | Numdam | MR 343661 | Zbl 0302.65087

[10] I. Demirdžić, v. Lilek and M. Perić, A collocated finite volume method for predicting flows at all speeds. Int. J. Numer. Methods Fluids 16 (1993) 1029-1050. | Zbl 0774.76066

[11] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in vol. VII of Handb. Numer. Anal. Edited by P. Ciarlet and J. Lions. North Holland (2000) 713-1020. | Zbl 0981.65095

[12] R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, Convergence of the MAC scheme for the compressible Stokes equations. SIAM J. Numer. Anal. 48 (2010) 2218-2246. | MR 2763662 | Zbl pre05931236

[13] E. Feireisl, Dynamics of Viscous Compressible Flows. In vol. 26 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2004). | MR 2040667 | Zbl 1080.76001

[14] T. Gallouët, L. Gastaldo, R. Herbin and J.-C. Latché, An unconditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations. Math. Model. Numer. Anal. 42 (2008) 303-331. | Numdam | MR 2405150 | Zbl 1132.35433

[15] 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 Vol. 2, − Prague, Czech Republic (2011) 39-56. | MR 2882362 | Zbl 1246.76094

[16] L. Gastaldo, R. Herbin and J.-C. Latché, A discretization of phase mass balance in fractional step algorithms for the drift-flux model. IMA J. Numer. Anal. 3 (2011) 116-146. | MR 2755939 | Zbl pre05853329

[17] L. Gastaldo, R. Herbin, J.-C. Latché and N. Therme, Explicit high order staggered schemes for the Euler equations (2014).

[18] D. Grapsas, R. Herbin, W. Kheriji and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible Navier-Stokes equations. Submitted (2014). | Numdam | Zbl 1132.35433

[19] J. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011-6045. | MR 2250931 | Zbl 1122.76072

[20] J. Guermond and R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C.R. Acad. Sci. Paris - Série I - Analyse Numérique 346 (2008) 801-806. | MR 2427085 | Zbl 1145.65079

[21] J. Guermond, R. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230 (2011) 4248-4267. | MR 2787948 | Zbl 1220.65134

[22] J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (2000) 167-188. | MR 1795396 | Zbl 0994.76051

[23] F. Harlow and A. Amsden, Numerical calculation of almost incompressible flow. J. Comput. Phys. 3 (1968) 80-93. | Zbl 0172.52903

[24] F. Harlow and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys. 8 (1971) 197-213. | Zbl 0221.76011

[25] F. Harlow and J. Welsh, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8 (1965) 2182-2189. | Zbl 1180.76043

[26] R. Herbin, W. Kheriji and J.-C. Latché, Staggered schemes for all speed flows. ESAIM Proc. 35 (2012) 22-150. | MR 3040778 | Zbl pre06023190

[27] R. Herbin, W. Kheriji and J.-C. Latché, Pressure correction staggered schemes for barotropic monophasic and two-phase flows. Comput. Fluids 88 (2013) 524-542. | MR 3131211

[28] R. Herbin and J.-C. Latché, Kinetic energy control in the MAC discretization of the compressible Navier-Stokes equations. Int. J. Finites Volumes 7 (2010). | MR 2753586

[29] R. Herbin, J.-C. Latché and K. Mallem, Convergence of the MAC scheme for the steady-state incompressible Navier-Stokes equations on non-uniform grids. Proc. of Finite Volumes for Complex Applications VII − Problems and Perspectives, Berlin, Germany (2014). | MR 3213365 | Zbl pre06362799

[30] R. Herbin, J.-C. Latché and T. Nguyen, An explicit staggered scheme for the shallow water and Euler equations. Submitted (2013).

[31] R. Herbin, J.-C. Latché and T. Nguyen, Explicit staggered schemes for the compressible euler equations. ESAIM Proc. 40 (2013) 83-102. | MR 3095649

[32] B. Hjertager, Computer simulation of reactive gas dynamics. Vol. 5 of Modeling, Identification and Control (1985) 211-236.

[33] Y. Hou and K. Mahesh, A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows. J. Comput. Phys. 205 (2005) 205-221. | Zbl 1088.76020

[34] R. Issa, Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys. 62 (1985) 40-65. | MR 825890 | Zbl 0619.76024

[35] R. Issa, A. Gosman and A. Watkins, The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comput. Phys. 62 (1986) 66-82. | MR 825891 | Zbl 0575.76008

[36] R. Issa and M. Javareshkian, Pressure-based compressible calculation method utilizing total variation diminishing schemes. AIAA J. 36 (1998) 1652-1657.

[37] S. Kadioglu, M. Sussman, S. Osher, J. Wright and M. Kang, A second order primitive preconditioner for solving all speed multi-phase flows. J. Comput. Phys. 209 (2005) 477-503. | MR 2151993 | Zbl 1138.76414

[38] K. Karki and S. Patankar, Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27 (1989) 1167-1174.

[39] M. Kobayashi and J. Pereira. Characteristic-based pressure correction at all speeds. AIAA J. 34 (1996) 272-280. | Zbl 0895.76054

[40] A. Kurganov and Y. Liu, New adaptative artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231 (2012) 8114-8132. | MR 2979844 | Zbl 1284.65112

[41] N. Kwatra, J. Su, J. Grétarsson and R. Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow. J. Comput. Phys. 228 (2009) 4146-4161. | MR 2524514 | Zbl 1273.76356

[42] J.-C. Latché and K. Saleh, A convergent staggered scheme for variable density incompressible Navier-Stokes equations. Submitted (2014).

[43] F.-S. Lien, A pressure-based unstructured grid method for all-speed flows. Int. J. Numer. Methods Fluids 33 (2000) 355-374. | Zbl 0977.76057

[44] P.-L. Lions, Mathematical Topics in Fluid Mechanics - Volume 2 - Compressible Models. Vol. 10 of Oxford Lect. Ser. Math. Appl. Oxford University Press (1998). | MR 1637634 | Zbl 1264.76002

[45] A. Majda and J. Sethian. The derivation and numerical solution of the equations for zero Mach number solution. Combust. Sci. Techn. 42 (1985) 185-205.

[46] R. Martineau and R. Berry, The pressure-corrected ICE finite element method for compressible flows on unstructured meshes. J. Comput. Phys. 198 (2004) 659-685. | Zbl 1116.76388

[47] J. Mcguirk and G. Page, Shock capturing using a pressure-correction method. AIAA J. 28 (1990) 1751-1757.

[48] F. Moukalled and M. Darwish, A high-resolution pressure-based algorithm for fluid flow at all speeds. J. Comput. Phys. 168 (2001) 101-133. | MR 1826910 | Zbl 0991.76047

[49] V. Moureau, C. Bérat and H. Pitsch, An efficient semi-implicit compressible solver for large-eddy simulations. J. Comput. Phys. 226 (2007) 1256-1270. | MR 2356372 | Zbl 1173.76321

[50] K. Nerinckx, J. Vierendeels and E. Dick, Mach-uniformity through the coupled pressure and temperature correction algorithm. J. Comput. Phys. 206 (2005) 597-623. | Zbl 1120.76300

[51] K. Nerinckx, J. Vierendeels and E. Dick. A Mach-uniform algorithm: coupled versus segregated approach. J. Comput. Phys. 224 (2007) 314-331. | MR 2322273 | Zbl 1261.76022

[52] P. Nithiarasu, R. Codina and O. Zienkiewicz, The Characteristic-Based Split (CBS) scheme - A unified approach to fluid dynamics. Int. J. Numer. Methods Engrg. 66 (2006) 1514-1546. | MR 2230959 | Zbl 1110.76324

[53] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Vol. 27 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2004). | Zbl 1088.35051

[54] G. Patnaik, R. Guirguis, J. Boris and E. Oran, A barely implicit correction for flux-corrected transport. J. Comput. Phys. 71 (1987) 1-20. | Zbl 0613.76077

[55] PELICANS, Collaborative development environment. Available on https://gforge.irsn.fr/gf/project/pelicans.

[56] L. Piar, F. Babik, R. Herbin and J.-C. Latché, A formally second order cell centered scheme for convection-diffusion equations on unstructured nonconforming grids. Int. J. Numer. Methods Fluids 71 (2013) 873-890. | MR 3019178

[57] E. Politis and K. Giannakoglou, A pressure-based algorithm for high-speed turbomachinery flows. Int. J. Numer. Methods Fluids 25 (1997) 63-80. | Zbl 0882.76057

[58] R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (1992) 97-111. | MR 1148797 | Zbl 0742.76051

[59] E. Sewall and D. Tafti, A time-accurate variable property algorithm for calculating flows with large temperature variations. Comput. Fluids 37 (2008) 51-63. | Zbl 1194.76185

[60] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Rat. Mech. Anal. 33 (1969) 377-385. | MR 244654 | Zbl 0207.16904

[61] S. Thakur and J. Wright, A multiblock operator-splitting algorithm for unsteady flows at all speeds in complex geometries. Int. J. Numer. Methods Fluids 46 (2004) 383-413. | MR 2087850 | Zbl 1112.76055

[62] N. Therme and Z. Chady, Comparison of consistent explicit schemes on staggered and colocated meshes (2014).

[63] E. Toro, Riemann solvers and numerical methods for fluid dynamics - A practical introduction, 3rd edition. Springer (2009). | MR 2731357 | Zbl 0923.76004

[64] D. Van Der Heul, C. Vuik and P. Wesseling, Stability analysis of segregated solution methods for compressible flow. Appl. Numer. Math. 38 (2001) 257-274. | MR 1847066 | Zbl 1017.76065

[65] D. Van Der Heul, C. Vuik and P. Wesseling. A conservative pressure-correction method for flow at all speeds. Comput. Fluids 32 (2003) 1113-1132. | MR 1966263 | Zbl 1046.76033

[66] J. Van Dormaal, G. Raithby and B. Mcdonald, The segregated approach to predicting viscous compressible fluid flows. Trans. ASME 109 (1987) 268-277.

[67] D. Vidović, A. Segal and P. Wesseling, A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys. 217 (2006) 277-294. | MR 2260602 | Zbl 1101.76037

[68] C. Wall, C. Pierce and P. Moin, A semi-implicit method for resolution of acoustic waves in low Mach number flows. J. Comput. Phys. 181 (2002) 545-563. | MR 1927401 | Zbl 1178.76264

[69] I. Wenneker, A. Segal and P. Wesseling, A Mach-uniform unstructured staggered grid method. Int. J. Numer. Methods Fluids 40 (2002) 1209-1235. | MR 1939062 | Zbl 1025.76023

[70] C. Xisto, J. Páscoa, P. Oliveira and D. Nicolini, A hybrid pressure-density-based algorithm for the Euler equations at all Mach number regimes. Int. J. Numer. Methods Fluids, online (2011).

[71] S. Yoon and T. Yabe, The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method. Comput. Phys. Commun. 119 (1999) 149-158. | Zbl 1175.76119

[72] O. Zienkiewicz and R. Codina, A general algorithm for compressible and incompressible flow - Part I. The split characteristic-based scheme. Int. J. Numer. Methods Fluids 20 (1995) 869-885. | MR 1333910 | Zbl 0837.76043