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Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension
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 |
  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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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