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Discontinuous Galerkin spectral/$$hp$$ element modelling of dispersive shallow water systems. (English) Zbl 1067.76057
From the summary: We consider the unstructured spectral/$$hp$$ discontinuous Galerkin formulation of weakly nonlinear dispersive Boussinesq equations and nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third-order mixed derivatives. To efficiently handle these high-order terms, we present a new scalar formulation based on the divergence of the momentum equations. Numerical computations illustrate the exponential convergence with regard to expansion order. Finally, we simulate solitary wave solutions.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76M22 Spectral methods applied to problems in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B25 Solitary waves for incompressible inviscid fluids
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##### References:
 [10] Dupont, F. (2001). Comparison of Numerical Methods for Modelling Ocean Circulation in Basins with Irregular Coasts, Ph.D. Thesis, McGill University. [13] Eskilsson, C., and Sherwin, S. J. A triangular spectral/hp discontinuous Galerkin method for modelling two-dimensional shallow water equations. Int. J. Numeric. Meth. Fluids, in press. · Zbl 1085.76544 [14] Giraldo, F. X. (1998). A spectral element semi-Lagrangian method for the shallow water equations on unstructured grids. Proceeding of the Fourth World Congress on Computational Mechanics.
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