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

R. Herbin; W. Kheriji; J.-C. Latché

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 6, page 1807-1857
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Herbin, R., Kheriji, W., and 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 48.6 (2014): 1807-1857. <http://eudml.org/doc/273331>.

@article{Herbin2014,
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.},
author = {Herbin, R., Kheriji, W., Latché, J.-C.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite volumes; finite elements; staggered; pressure correction; Euler equations; shallow-water equations; compressible flows; analysis},
language = {eng},
number = {6},
pages = {1807-1857},
publisher = {EDP-Sciences},
title = {On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations},
url = {http://eudml.org/doc/273331},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Herbin, R.
AU - Kheriji, W.
AU - Latché, J.-C.
TI - On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 6
SP - 1807
EP - 1857
AB - 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.
LA - eng
KW - finite volumes; finite elements; staggered; pressure correction; Euler equations; shallow-water equations; compressible flows; analysis
UR - http://eudml.org/doc/273331
ER -

References

top
  1. [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 Fluids66 (2011) 555–580. Zbl1321.76035MR2839213
  2. [2] F. Archambeau, J.-M. Hérard and J. Laviéville, Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases. Comput. Fluids38 (2009) 1495–1509. Zbl1242.76152MR2645756
  3. [3] R. Berry, Notes on PCICE method: simplification, generalization and compressibility properties. J. Comput. Phys.215 (2006) 6–11. Zbl1140.76412MR2215649
  4. [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. Zbl0918.76054MR1619651
  5. [5] CALIF3S. A software components library for the computation of reactive turbulent flows. Available on https://gforge.irsn.fr/gf/project/isis. 
  6. [6] V. Casulli and D. Greenspan, Pressure method for the numerical solution of transient, compressible fluid flows. Int. J. Numer. Methods Fluids4 (1984) 1001–1012. Zbl0549.76050
  7. [7] A. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput.22 (1968) 745–762. Zbl0198.50103MR242392
  8. [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. Zbl0875.65086MR1115237
  9. [9] M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Série Rouge7 (1973) 33–75. Zbl0302.65087MR343661
  10. [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. Zbl0774.76066
  11. [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. Zbl0981.65095
  12. [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. Zbl05931236MR2763662
  13. [13] E. Feireisl, Dynamics of Viscous Compressible Flows. In vol. 26 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2004). Zbl1080.76001MR2040667
  14. [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. Zbl1132.35433MR2405150
  15. [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. Zbl1246.76094MR2882362
  16. [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. Zbl05853329MR2755939
  17. [17] L. Gastaldo, R. Herbin, J.-C. Latché and N. Therme, Explicit high order staggered schemes for the Euler equations (2014). 
  18. [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). Zbl1132.35433
  19. [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. Zbl1122.76072MR2250931
  20. [20] J. Guermond and R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C.R. Acad. Sci. Paris – Série I – Analyse Numérique346 (2008) 801–806. Zbl1145.65079MR2427085
  21. [21] J. Guermond, R. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys.230 (2011) 4248–4267. Zbl1220.65134MR2787948
  22. [22] J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows. J. Comput. Phys.165 (2000) 167–188. Zbl0994.76051MR1795396
  23. [23] F. Harlow and A. Amsden, Numerical calculation of almost incompressible flow. J. Comput. Phys.3 (1968) 80–93. Zbl0172.52903
  24. [24] F. Harlow and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys.8 (1971) 197–213. Zbl0221.76011
  25. [25] F. Harlow and J. Welsh, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids8 (1965) 2182–2189. Zbl1180.76043
  26. [26] R. Herbin, W. Kheriji and J.-C. Latché, Staggered schemes for all speed flows. ESAIM Proc.35 (2012) 22–150. Zbl06023190MR3040778
  27. [27] R. Herbin, W. Kheriji and J.-C. Latché, Pressure correction staggered schemes for barotropic monophasic and two-phase flows. Comput. Fluids88 (2013) 524–542. MR3131211
  28. [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). Zbl05931236MR2753586
  29. [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). Zbl1304.76038MR3213365
  30. [30] R. Herbin, J.-C. Latché and T. Nguyen, An explicit staggered scheme for the shallow water and Euler equations. Submitted (2013). Zbl1329.76206
  31. [31] R. Herbin, J.-C. Latché and T. Nguyen, Explicit staggered schemes for the compressible euler equations. ESAIM Proc.40 (2013) 83–102. Zbl1329.76206MR3095649
  32. [32] B. Hjertager, Computer simulation of reactive gas dynamics. Vol. 5 of Modeling, Identification and Control (1985) 211–236. 
  33. [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. Zbl1088.76020
  34. [34] R. Issa, Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys.62 (1985) 40–65. Zbl0619.76024MR825890
  35. [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. Zbl0575.76008MR825891
  36. [36] R. Issa and M. Javareshkian, Pressure-based compressible calculation method utilizing total variation diminishing schemes. AIAA J.36 (1998) 1652–1657. 
  37. [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. Zbl1138.76414MR2151993
  38. [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. [39] M. Kobayashi and J. Pereira. Characteristic-based pressure correction at all speeds. AIAA J.34 (1996) 272–280. Zbl0895.76054
  40. [40] A. Kurganov and Y. Liu, New adaptative artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys.231 (2012) 8114–8132. Zbl1284.65112MR2979844
  41. [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. Zbl1273.76356MR2524514
  42. [42] J.-C. Latché and K. Saleh, A convergent staggered scheme for variable density incompressible Navier-Stokes equations. Submitted (2014). Zbl1311.35189
  43. [43] F.-S. Lien, A pressure-based unstructured grid method for all-speed flows. Int. J. Numer. Methods Fluids33 (2000) 355–374. Zbl0977.76057
  44. [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). Zbl1264.76002MR1637634
  45. [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. [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. Zbl1116.76388
  47. [47] J. McGuirk and G. Page, Shock capturing using a pressure-correction method. AIAA J.28 (1990) 1751–1757. 
  48. [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. Zbl0991.76047MR1826910
  49. [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. Zbl1173.76321MR2356372
  50. [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. Zbl1120.76300
  51. [51] K. Nerinckx, J. Vierendeels and E. Dick. A Mach-uniform algorithm: coupled versus segregated approach. J. Comput. Phys.224 (2007) 314–331. Zbl1261.76022MR2322273
  52. [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. Zbl1110.76324MR2230959
  53. [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). Zbl1088.35051
  54. [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. Zbl0613.76077
  55. [55] PELICANS, Collaborative development environment. Available on https://gforge.irsn.fr/gf/project/pelicans. 
  56. [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 Fluids71 (2013) 873–890. MR3019178
  57. [57] E. Politis and K. Giannakoglou, A pressure-based algorithm for high-speed turbomachinery flows. Int. J. Numer. Methods Fluids25 (1997) 63–80. Zbl0882.76057
  58. [58] R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ.8 (1992) 97–111. Zbl0742.76051MR1148797
  59. [59] E. Sewall and D. Tafti, A time-accurate variable property algorithm for calculating flows with large temperature variations. Comput. Fluids37 (2008) 51–63. Zbl1194.76185
  60. [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. Zbl0207.16904MR244654
  61. [61] S. Thakur and J. Wright, A multiblock operator-splitting algorithm for unsteady flows at all speeds in complex geometries. Int. J. Numer. Methods Fluids46 (2004) 383–413. Zbl1112.76055MR2087850
  62. [62] N. Therme and Z. Chady, Comparison of consistent explicit schemes on staggered and colocated meshes (2014). 
  63. [63] E. Toro, Riemann solvers and numerical methods for fluid dynamics – A practical introduction, 3rd edition. Springer (2009). Zbl0923.76004MR2731357
  64. [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. Zbl1017.76065MR1847066
  65. [65] D. Van der Heul, C. Vuik and P. Wesseling. A conservative pressure-correction method for flow at all speeds. Comput. Fluids32 (2003) 1113–1132. Zbl1046.76033MR1966263
  66. [66] J. Van Dormaal, G. Raithby and B. McDonald, The segregated approach to predicting viscous compressible fluid flows. Trans. ASME109 (1987) 268–277. 
  67. [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. Zbl1101.76037MR2260602
  68. [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. Zbl1178.76264MR1927401
  69. [69] I. Wenneker, A. Segal and P. Wesseling, A Mach-uniform unstructured staggered grid method. Int. J. Numer. Methods Fluids40 (2002) 1209–1235. Zbl1025.76023MR1939062
  70. [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. [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. Zbl1175.76119
  72. [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. Zbl0837.76043MR1333910

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