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Spectral/\(hp\) discontinuous Galerkin methods for modelling 2D Boussinesq equations. (English) Zbl 1084.76058
Summary: We present spectral/\(hp\) discontinuous Galerkin methods for modelling weakly nonlinear and dispersive water waves, described by a set of depth-integrated Boussinesq equations, on unstructured triangular meshes. When solving the equations, two different formulations are considered: directly solving the coupled momentum equations and the ’scalar method’, in which a wave continuity equation is solved as an intermediate step. We demonstrate that the approaches are fully equivalent and give identical results in terms of accuracy, convergence and restriction on the time step. However, the scalar method is shown to be more CPU efficient for high-order expansions, in addition to requiring less storage.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Software:
UMFPACK
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