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A nonlinear Galerkin method for the shallow-water equations on periodic domains. (English) Zbl 1002.76087
Summary: A nonlinear Galerkin method for the shallow-water equations is developed based on spectral transforms. The scheme is compared to a pseudo-spectral Galerkin method. Numerical results indicate that the nonlinear scheme has the potential advantage of providing similar accuracy at a lower cost than Galerkin method. The nonlinear method has also less restrictive stability conditions.

76M22 Spectral methods applied to problems in fluid mechanics
76D33 Waves for incompressible viscous fluids
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Canuto, C.; Husaini, Y.; Quarteroni, A.; Zang, T., Spectral methods in fluid dynamics, (1988)
[2] Cardenas, J.W., Variedades inerciais aproximadas e Métodos de Galerkin não linear para as equações de água rasa (approximate inertial manifolds and non-linear Galerkin methods for the shallow-water equations), (1999), Universidade de São Paulo S. Paulo
[3] V. Casulli, and, E. Cattani, Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl, 27, 99, 1994. · Zbl 0796.76052
[4] A. Debussche, T. Dubois, and, R. Temam, 1993, The nonlinear Galerkin method: A multi-scale method applied to the simulation of homogeneos turbulent flows, Icase Report No. 93-93.
[5] L.; Dettori; Gottlieb, D.; Temam, R., A nonlinear Galerkin method: the two-level Fourier-collocation case, J. sci. comput., 10, 371, (1995) · Zbl 0859.65111
[6] di Martino, B.; Orenga, P., Résolution des équations de shallow water par la méthode de Galerkin non linéaire, M^{2} AN, Math. model. numer. anal., 32, 451, (1998) · Zbl 0916.76031
[7] Dubois, T.; Jabertau, F.; Temam, R., Solution of the incompressible navier – stokes equations by the nonlinear Galerkin methods, J. sci. comput., 8, 167, (1993) · Zbl 0783.76068
[8] E. Eliasen, B. Machenhauer, and, E. Rasmussen, On a numerical method for integration of the hydrodynamical equations with a spectral representation of the horizontal fields, Report Nr. 2, Institut for Teoretisk Meteorologi, Københavns Universitet, Haraldsgade 6, DK-2200, Copenhagen N, Denmark, 1970.
[9] Foias, C.; Manley, O.P.; Temam, R., Modelization of the iteration of small and large eddies in two dimensional turbulent flows M^{2} AN, Math. model. numer. anal., 22, 93, (1988) · Zbl 0663.76054
[10] Gottileb, D.; Temam, R., Implementation of the nonlinear Galerkin method with pseudospectral (collocation) discretizations, Appl. numer. math., 12, 119, (1993) · Zbl 0782.65122
[11] Jabertau, F.; Rosier, C.; Temam, R., A nonlinear Galerkin method for the Navier-Stokes equation, Computer meth. appl. mech. eng., 80, 245, (1990) · Zbl 0722.76039
[12] Jones, D.E.; Margolin, L.G.; Titi, E.S., On the effectiveness of the approximate inertial manifold—a computational study, Thoret. comput. fluid dynamics, 7, 243, (1995) · Zbl 0838.76066
[13] Lorenz, E., Attractor sets and quasi-geostrophic equilibrium, J. atmos. sci., 37, 1685, (1980)
[14] Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM J. num. anal., 26, 1139, (1989) · Zbl 0683.65083
[15] Orzag, S.A., Transform method for calculation of vector – coupled sums: application to the spectral form of the vorticity equation, J. atmos. sci., 27, 890, (1970)
[16] Ritchie, H.; Temperton, C.; Simmons, A.J.; Hortal, M.; Davies, T.; Dent, D.; Hamrud, M., Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF foracast model, Mon. wea. rev., 123, 489, (1995)
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