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On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

Published online by Cambridge University Press:  10 October 2014

R. Herbin ,
W. Kheriji  and
J.-C. Latché
R. Herbin
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France.. raphaele.herbin@univ-amu.fr
W. Kheriji
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), BP 3, 13115 Saint-Paul-lez-Durance cedex, France. ; kheriji.walid@gmail.com; jean-claude.latche@irsn.fr
J.-C. Latché
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), BP 3, 13115 Saint-Paul-lez-Durance cedex, France. ; kheriji.walid@gmail.com; jean-claude.latche@irsn.fr
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Abstract

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.

Keywords

Finite volumesfinite elementsstaggeredpressure correctionEuler equationsshallow-water equationscompressible flowsanalysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1807 - 1857
Copyright
© EDP Sciences, SMAI 2014

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References

Ansanay-Alex, G., Babik, F., Latché, J.-C. and Vola, D., 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. Google Scholar
Archambeau, F., Hérard, J.-M. and Laviéville, J., Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases. Comput. Fluids 38 (2009) 1495–1509. Google Scholar
Berry, R., Notes on PCICE method: simplification, generalization and compressibility properties. J. Comput. Phys. 215 (2006) 6–11. Google Scholar
Bijl, H. and Wesseling, P., A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141 (1998) 153–173. Google Scholar
CALIF3S. A software components library for the computation of reactive turbulent flows. Available on https://gforge.irsn.fr/gf/project/isis.
Casulli, V. and Greenspan, D., Pressure method for the numerical solution of transient, compressible fluid flows. Int. J. Numer. Methods Fluids 4 (1984) 1001–1012. Google Scholar
Chorin, A., Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. Google Scholar
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.
Crouzeix, M. and Raviart, P., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Série Rouge 7 (1973) 33–75. Google Scholar
Demirdžić, I., Lilek, v. and Perić, M., A collocated finite volume method for predicting flows at all speeds. Int. J. Numer. Methods Fluids 16 (1993) 1029–1050. Google Scholar
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.
Eymard, R., Gallouët, T., Herbin, R. and Latché, J.-C., Convergence of the MAC scheme for the compressible Stokes equations. SIAM J. Numer. Anal. 48 (2010) 2218–2246. Google Scholar
E. Feireisl, Dynamics of Viscous Compressible Flows. In vol. 26 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2004).
Gallouët, T., Gastaldo, L., Herbin, R. and Latché, J.-C., An unconditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations. Math. Model. Numer. Anal. 42 (2008) 303–331. Google Scholar
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.
Gastaldo, L., Herbin, R. and Latché, J.-C., A discretization of phase mass balance in fractional step algorithms for the drift-flux model. IMA J. Numer. Anal. 3 (2011) 116–146. Google Scholar
L. Gastaldo, R. Herbin, J.-C. Latché and N. Therme, Explicit high order staggered schemes for the Euler equations (2014).
D. Grapsas, R. Herbin, W. Kheriji and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible Navier-Stokes equations. Submitted (2014).
Guermond, J., Minev, P. and Shen, J., An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011–6045. Google Scholar
Guermond, J. and Pasquetti, R., Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C.R. Acad. Sci. Paris – Série I – Analyse Numérique 346 (2008) 801–806. Google Scholar
Guermond, J., Pasquetti, R. and Popov, B., Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230 (2011) 4248–4267. Google Scholar
Guermond, J.-L. and Quartapelle, L., A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (2000) 167–188. Google Scholar
Harlow, F. and Amsden, A., Numerical calculation of almost incompressible flow. J. Comput. Phys. 3 (1968) 80–93. Google Scholar
Harlow, F. and Amsden, A., A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys. 8 (1971) 197–213. Google Scholar
Harlow, F. and Welsh, J., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8 (1965) 2182–2189. Google Scholar
Herbin, R., Kheriji, W. and Latché, J.-C., Staggered schemes for all speed flows. ESAIM Proc. 35 (2012) 22–150. Google Scholar
Herbin, R., Kheriji, W. and Latché, J.-C., Pressure correction staggered schemes for barotropic monophasic and two-phase flows. Comput. Fluids 88 (2013) 524–542. Google Scholar
Herbin, R. and Latché, J.-C., Kinetic energy control in the MAC discretization of the compressible Navier-Stokes equations. Int. J. Finites Volumes 7 (2010). Google Scholar
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).
R. Herbin, J.-C. Latché and T. Nguyen, An explicit staggered scheme for the shallow water and Euler equations. Submitted (2013).
Herbin, R., Latché, J.-C. and Nguyen, T., Explicit staggered schemes for the compressible euler equations. ESAIM Proc. 40 (2013) 83–102. Google Scholar
B. Hjertager, Computer simulation of reactive gas dynamics. Vol. 5 of Modeling, Identification and Control (1985) 211–236.
Hou, Y. and Mahesh, K., A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows. J. Comput. Phys. 205 (2005) 205–221. Google Scholar
Issa, R., Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys. 62 (1985) 40–65. Google Scholar
Issa, R., Gosman, A. and Watkins, A., The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comput. Phys. 62 (1986) 66–82. Google Scholar
Issa, R. and Javareshkian, M., Pressure-based compressible calculation method utilizing total variation diminishing schemes. AIAA J. 36 (1998) 1652–1657. Google Scholar
Kadioglu, S., Sussman, M., Osher, S., Wright, J. and Kang, M., A second order primitive preconditioner for solving all speed multi-phase flows. J. Comput. Phys. 209 (2005) 477–503. Google Scholar
Karki, K. and Patankar, S., Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27 (1989) 1167–1174. Google Scholar
Kobayashi, M. and Pereira, J.. Characteristic-based pressure correction at all speeds. AIAA J. 34 (1996) 272–280. Google Scholar
Kurganov, A. and Liu, Y., New adaptative artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231 (2012) 8114–8132. Google Scholar
Kwatra, N., Su, J., Grétarsson, J. and Fedkiw, R., A method for avoiding the acoustic time step restriction in compressible flow. J. Comput. Phys. 228 (2009) 4146–4161. Google Scholar
J.-C. Latché and K. Saleh, A convergent staggered scheme for variable density incompressible Navier-Stokes equations. Submitted (2014).
Lien, F.-S., A pressure-based unstructured grid method for all-speed flows. Int. J. Numer. Methods Fluids 33 (2000) 355–374. Google Scholar
P.-L. Lions, Mathematical Topics in Fluid Mechanics – Volume 2 – Compressible Models. Vol. 10 of Oxford Lect. Ser. Math. Appl. Oxford University Press (1998).
Majda, A. and Sethian, J.. The derivation and numerical solution of the equations for zero Mach number solution. Combust. Sci. Techn. 42 (1985) 185–205. Google Scholar
Martineau, R. and Berry, R., The pressure-corrected ICE finite element method for compressible flows on unstructured meshes. J. Comput. Phys. 198 (2004) 659–685. Google Scholar
McGuirk, J. and Page, G., Shock capturing using a pressure-correction method. AIAA J. 28 (1990) 1751–1757. Google Scholar
Moukalled, F. and Darwish, M., A high-resolution pressure-based algorithm for fluid flow at all speeds. J. Comput. Phys. 168 (2001) 101–133. Google Scholar
Moureau, V., Bérat, C. and Pitsch, H., An efficient semi-implicit compressible solver for large-eddy simulations. J. Comput. Phys. 226 (2007) 1256–1270. Google Scholar
Nerinckx, K., Vierendeels, J. and Dick, E., Mach-uniformity through the coupled pressure and temperature correction algorithm. J. Comput. Phys. 206 (2005) 597–623. Google Scholar
Nerinckx, K., Vierendeels, J. and Dick, E.. A Mach-uniform algorithm: coupled versus segregated approach. J. Comput. Phys. 224 (2007) 314–331. Google Scholar
Nithiarasu, P., Codina, R. and Zienkiewicz, O., The Characteristic-Based Split (CBS) scheme – A unified approach to fluid dynamics. Int. J. Numer. Methods Engrg. 66 (2006) 1514–1546. Google Scholar
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).
Patnaik, G., Guirguis, R., Boris, J. and Oran, E., A barely implicit correction for flux-corrected transport. J. Comput. Phys. 71 (1987) 1–20. Google Scholar
PELICANS, Collaborative development environment. Available on https://gforge.irsn.fr/gf/project/pelicans.
Piar, L., Babik, F., Herbin, R. and Latché, J.-C., A formally second order cell centered scheme for convection-diffusion equations on unstructured nonconforming grids. Int. J. Numer. Methods Fluids 71 (2013) 873–890. Google Scholar
Politis, E. and Giannakoglou, K., A pressure-based algorithm for high-speed turbomachinery flows. Int. J. Numer. Methods Fluids 25 (1997) 63–80. Google Scholar
Rannacher, R. and Turek, S.. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (1992) 97–111. Google Scholar
Sewall, E. and Tafti, D., A time-accurate variable property algorithm for calculating flows with large temperature variations. Comput. Fluids 37 (2008) 51–63. Google Scholar
Temam, R., 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. Google Scholar
Thakur, S. and Wright, J., A multiblock operator-splitting algorithm for unsteady flows at all speeds in complex geometries. Int. J. Numer. Methods Fluids 46 (2004) 383–413. Google Scholar
N. Therme and Z. Chady, Comparison of consistent explicit schemes on staggered and colocated meshes (2014).
E. Toro, Riemann solvers and numerical methods for fluid dynamics – A practical introduction, 3rd edition. Springer (2009).
Van der Heul, D., Vuik, C. and Wesseling, P., Stability analysis of segregated solution methods for compressible flow. Appl. Numer. Math. 38 (2001) 257–274. Google Scholar
Van der Heul, D., Vuik, C. and Wesseling, P.. A conservative pressure-correction method for flow at all speeds. Comput. Fluids 32 (2003) 1113–1132.
Van Dormaal, J., Raithby, G. and McDonald, B., The segregated approach to predicting viscous compressible fluid flows. Trans. ASME 109 (1987) 268–277. Google Scholar
Vidović, D., Segal, A. and Wesseling, P., A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys. 217 (2006) 277–294. Google Scholar
Wall, C., Pierce, C. and Moin, P., A semi-implicit method for resolution of acoustic waves in low Mach number flows. J. Comput. Phys. 181 (2002) 545–563. Google Scholar
Wenneker, I., Segal, A. and Wesseling, P., A Mach-uniform unstructured staggered grid method. Int. J. Numer. Methods Fluids 40 (2002) 1209–1235. Google Scholar
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).
Yoon, S. and Yabe, T., 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. Google Scholar
Zienkiewicz, O. and Codina, R., A general algorithm for compressible and incompressible flow – Part I. The split characteristic-based scheme. Int. J. Numer. Methods Fluids 20 (1995) 869–885.Google Scholar