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

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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

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

Abstract
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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)*, (1995), 367. 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. 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)*, 14 (2009), 577. 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, 14 (2008), 481. doi: 10.1016/S1570-8659(08)00211-1. Google Scholar

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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)*, (1995), 367. 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. 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)*, 14 (2009), 577. 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, 14 (2008), 481. doi: 10.1016/S1570-8659(08)00211-1. Google Scholar

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