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October  2016, 9(5): 1565-1574. doi: 10.3934/dcdss.2016063

About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows

Eric Blayo 1, and  Antoine Rousseau 2,

1. 

Univ. Grenoble Alpes, LJK, BP 53, 38041 Grenoble cedex 9, France
2. 

Inria and Institut Montpelliérain Alexander Grothendieck, Team LEM0N, Bat 5 - CC05 017, 860 rue Saint-Priest, 34095 Montpellier Cedex 5, France

Received  June 2015 Revised  August 2015 Published  October 2016

In this work we are interested in the search of interface conditions to couple hydrostatic and nonhydrostatic ocean models. To this aim, we consider simplified systems and use a time discretization to handle linear equations. We recall the links between the two models (with the particular role of the aspect ratio $\delta=H/L\ll 1$) and introduce an iterative method based on the Schwarz algorithm (widely used in domain decomposition methods).
    The convergence of this method depends strongly on the choice of interface conditions: this is why we look for exact absorbing conditions and their approximations in order to provide tractable and efficient coupling algorithms.
Keywords: Schwarz alternating method., coupling methods, optimized boundary conditions, Domain decomposition, hydrostatic approximation, Navier-Stokes equations.
Mathematics Subject Classification: Primary: 65N55, 76D0.
Citation: Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063
References:
[1]

E. Audusse, P. Dreyfuss and B. Merlet, Schwarz waveform relaxation for primitive equations of the ocean, SIAM J. Sci. Comput., 32 (2010), 2908-2936. doi: 10.1137/090770059.  Google Scholar

[2]

P. Azerad and F. Guillen, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluids dynamics, SIAM J. Math. Anal., 33 (2001), 847-859. doi: 10.1137/S0036141000375962.  Google Scholar

[3]

E. Blayo, D. Cherel and A. Rousseau, Towards optimized Schwarz methods for the Navier-Stokes equations, J. Sci. Comput., 66 (2016), 275-295. doi: 10.1007/s10915-015-0020-9.  Google Scholar

[4]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.  Google Scholar

[5]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4.  Google Scholar

[6]

O. B. Fringer, M. Gerritsen and R. Street, An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal-ocean simulator, Ocean Modell., 14 (2006), 139-173. doi: 10.1016/j.ocemod.2006.03.006.  Google Scholar

[7]

O. B. Fringer, J. C. McWilliams and R. L. Street, A new hybrid model for coastal simulations, Oceanography, 19 (2006), 46-59. doi: 10.5670/oceanog.2006.91.  Google Scholar

[8]

P. C. Gallacher, D. A. Hebert and M. R. Schaferkotter, Nesting a nonhydrostatic model in a hydrostatic model: The boundary interface, Ocean Modell., 40 (2011), 190-198. doi: 10.1016/j.ocemod.2011.08.006.  Google Scholar

[9]

M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255.  Google Scholar

[10]

G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020.  Google Scholar

[11]

J.-L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.  Google Scholar

[12]

C. Lucas and A. Rousseau, New developments and cosine effect in the viscous shallow water and quasi-geostrophic equations, Multiscale Model. Simul., 7 (2008), 796-813. doi: 10.1137/070705453.  Google Scholar

[13]

J. Marshall, A. Adcroft, C. Hill, L. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res. Ocean., 102 (1997), 5753-5766. doi: 10.1029/96JC02775.  Google Scholar

[14]

J. Marshall, C. Hill, L. Perelman and A. Adcroft, Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, J. Geophys. Res. Ocean., 102 (1997), 5733-5752. doi: 10.1029/96JC02776.  Google Scholar

[15]

F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models and Methods in Applied Sciences, 5 (1995), 67-93. doi: 10.1142/S021820259500005X.  Google Scholar

[16]

F. Nataf, F. Rogier and E. de Sturler, Domain decomposition methods for fluid dynamics, in Navier-Stokes Equations and Related Nonlinear Problems (ed. A. Sequeira), Springer US, (1995), 367-376.  Google Scholar

[17]

M. F. Peeters, W. G. Habashi and E. G. Dueck, Finite element streamfunction - vorticity solutions of the incompressible Navier-Stokes equations, Int. J. Num. Methods in Fluids, 7 (2008), 17-27. Google Scholar

[18]

M. Petcu, R. Temam and M. Ziane, Some mathematical problems in fluid dynamics, in Handbook of Numerical Analysis (eds. P.G. Ciarlet), Elsevier, 14 (2009), 577-750. doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar

[19]

A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domain, in Handbook of Numerical Analysis, (eds. P.G. Ciarlet), Elsevier, 14 (2008), 481-575. doi: 10.1016/S1570-8659(08)00211-1.  Google Scholar

show all references

References:
[1]

E. Audusse, P. Dreyfuss and B. Merlet, Schwarz waveform relaxation for primitive equations of the ocean, SIAM J. Sci. Comput., 32 (2010), 2908-2936. doi: 10.1137/090770059.  Google Scholar

[2]

P. Azerad and F. Guillen, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluids dynamics, SIAM J. Math. Anal., 33 (2001), 847-859. doi: 10.1137/S0036141000375962.  Google Scholar

[3]

E. Blayo, D. Cherel and A. Rousseau, Towards optimized Schwarz methods for the Navier-Stokes equations, J. Sci. Comput., 66 (2016), 275-295. doi: 10.1007/s10915-015-0020-9.  Google Scholar

[4]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.  Google Scholar

[5]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4.  Google Scholar

[6]

O. B. Fringer, M. Gerritsen and R. Street, An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal-ocean simulator, Ocean Modell., 14 (2006), 139-173. doi: 10.1016/j.ocemod.2006.03.006.  Google Scholar

[7]

O. B. Fringer, J. C. McWilliams and R. L. Street, A new hybrid model for coastal simulations, Oceanography, 19 (2006), 46-59. doi: 10.5670/oceanog.2006.91.  Google Scholar

[8]

P. C. Gallacher, D. A. Hebert and M. R. Schaferkotter, Nesting a nonhydrostatic model in a hydrostatic model: The boundary interface, Ocean Modell., 40 (2011), 190-198. doi: 10.1016/j.ocemod.2011.08.006.  Google Scholar

[9]

M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255.  Google Scholar

[10]

G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020.  Google Scholar

[11]

J.-L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.  Google Scholar

[12]

C. Lucas and A. Rousseau, New developments and cosine effect in the viscous shallow water and quasi-geostrophic equations, Multiscale Model. Simul., 7 (2008), 796-813. doi: 10.1137/070705453.  Google Scholar

[13]

J. Marshall, A. Adcroft, C. Hill, L. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res. Ocean., 102 (1997), 5753-5766. doi: 10.1029/96JC02775.  Google Scholar

[14]

J. Marshall, C. Hill, L. Perelman and A. Adcroft, Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, J. Geophys. Res. Ocean., 102 (1997), 5733-5752. doi: 10.1029/96JC02776.  Google Scholar

[15]

F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models and Methods in Applied Sciences, 5 (1995), 67-93. doi: 10.1142/S021820259500005X.  Google Scholar

[16]

F. Nataf, F. Rogier and E. de Sturler, Domain decomposition methods for fluid dynamics, in Navier-Stokes Equations and Related Nonlinear Problems (ed. A. Sequeira), Springer US, (1995), 367-376.  Google Scholar

[17]

M. F. Peeters, W. G. Habashi and E. G. Dueck, Finite element streamfunction - vorticity solutions of the incompressible Navier-Stokes equations, Int. J. Num. Methods in Fluids, 7 (2008), 17-27. Google Scholar

[18]

M. Petcu, R. Temam and M. Ziane, Some mathematical problems in fluid dynamics, in Handbook of Numerical Analysis (eds. P.G. Ciarlet), Elsevier, 14 (2009), 577-750. doi: 10.1016/S1570-8659(08)00212-3.  Google Scholar

[19]

A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domain, in Handbook of Numerical Analysis, (eds. P.G. Ciarlet), Elsevier, 14 (2008), 481-575. doi: 10.1016/S1570-8659(08)00211-1.  Google Scholar

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