On the two-dimensional compressible isentropic Navier–Stokes equations

Catherine Giacomoni; Pierre Orenga

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 6, page 1091-1109
  • ISSN: 0764-583X

Abstract

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We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with γ = c p / c v = 2 . These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions (1998). The second part is devoted to the numerical analysis and mainly deals with problems of geophysical fluids. We compare the simulations obtained with this compressible isentropic Navier–Stokes model and those obtained with a shallow water model (Di Martino et al., 1999). At first, the computations are executed on a simplified domain in order to validate the method by comparison with existing numerical results and then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analytical example presented by Weigant (1995) which shows that even if the data are rather smooth, we cannot have bounds on ρ in Lp for p large if γ < 2 when N=2.

How to cite

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Giacomoni, Catherine, and Orenga, Pierre. "On the two-dimensional compressible isentropic Navier–Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 36.6 (2010): 1091-1109. <http://eudml.org/doc/194141>.

@article{Giacomoni2010,
abstract = { We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma= \displaystyle\{\{c_\{p\}\}/\{c_\{v\}\}\}=2$. These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions (1998). The second part is devoted to the numerical analysis and mainly deals with problems of geophysical fluids. We compare the simulations obtained with this compressible isentropic Navier–Stokes model and those obtained with a shallow water model (Di Martino et al., 1999). At first, the computations are executed on a simplified domain in order to validate the method by comparison with existing numerical results and then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analytical example presented by Weigant (1995) which shows that even if the data are rather smooth, we cannot have bounds on ρ in Lp for p large if $\gamma<2$ when N=2. },
author = {Giacomoni, Catherine, Orenga, Pierre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Navier–Stokes; compressible; shallow water; time-discretisation; Galerkin.; time-discretized problem; Galerkin method; finite element Modulef software; square domain; shallow water model; Navier-Stokes equations},
language = {eng},
month = {3},
number = {6},
pages = {1091-1109},
publisher = {EDP Sciences},
title = {On the two-dimensional compressible isentropic Navier–Stokes equations},
url = {http://eudml.org/doc/194141},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Giacomoni, Catherine
AU - Orenga, Pierre
TI - On the two-dimensional compressible isentropic Navier–Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 6
SP - 1091
EP - 1109
AB - We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma= \displaystyle{{c_{p}}/{c_{v}}}=2$. These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions (1998). The second part is devoted to the numerical analysis and mainly deals with problems of geophysical fluids. We compare the simulations obtained with this compressible isentropic Navier–Stokes model and those obtained with a shallow water model (Di Martino et al., 1999). At first, the computations are executed on a simplified domain in order to validate the method by comparison with existing numerical results and then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analytical example presented by Weigant (1995) which shows that even if the data are rather smooth, we cannot have bounds on ρ in Lp for p large if $\gamma<2$ when N=2.
LA - eng
KW - Navier–Stokes; compressible; shallow water; time-discretisation; Galerkin.; time-discretized problem; Galerkin method; finite element Modulef software; square domain; shallow water model; Navier-Stokes equations
UR - http://eudml.org/doc/194141
ER -

References

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  2. R.J. DiPerna and P.L. Lions, On the cauchy problem for boltzman equations: global existence and weak stability. C.R. Acad. Sci. Paris Sér. I Math.306 (1988) 343-346.  
  3. R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and sobolev spaces. Invent. Math.98 (1989) 511-547.  
  4. V. Girault and P.A. Raviart, Finite Elements Methods of the Navier-Stokes Equations. Springer-Verlag (1986).  
  5. P.L. Lions, Mathematical Topics in Fluid Mechanics, Incompressible models. Vol. 1, Oxford Science Publications (1996).  
  6. P.L. Lions, Mathematical Topics in Fluid Mechanics, Compressible models. Vol. 2, Oxford Science Publications (1998).  
  7. B. Di Martino, F.J. Chatelon and P. Orenga, The nonlinear Galerkin's method applied to the shallow water equations. Math. Models Methods Appl. Sci.9 (1999) 825-854.  
  8. P. Orenga, Analyse de quelques problèmes d'océanographie physique. Ph.D. thesis, Université de Corse, Corte (1992).  
  9. P. Orenga, Construction d'une base spéciale pour la résolution de quelques problèmes non linéaires d'océanographie physique en dimension deux, in Nonlinear partial differential equations and their applications, D. Cioranescu and J.L. Lions, Vol. 13. Longman, Pitman Res. Notes Math. Ser. 391 (1998) 234-258.  
  10. V.A. Solonnikov, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 56 (1976) 128-142. English translation in J. Soviet Math. 14 (1980) 1120-1133.  
  11. V.A. Weigant, An exemple of non-existence globally in time of a solution of the Navier-Stokes equations for a compressible viscous barotropic fluid. Russian Acad. Sci. Doklady Mathematics50 (1995) 397-399.  
  12. E. Zeidler, Fixed-point theorems, in Nonlinear Functional Analysis and its Applications, Vol. 1, Springer-Verlag (1986).  

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