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A high-order triangular discontinuous Galerkin oceanic shallow water model. (English) Zbl 1290.86002
Summary: A high-order triangular discontinuous Galerkin (DG) method is applied to the two-dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order \(2N\) Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight-edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases - three of which have analytic solutions. The three tests having analytic solutions show that the high-order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered.

MSC:
86-08 Computational methods for problems pertaining to geophysics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
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