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Dynamic \(p\)-adaptive Runge-Kutta discontinuous Galerkin methods for the shallow water equations. (English) Zbl 1227.76032

Summary: In this paper, dynamic \(p\)-adaptive Runge-Kutta discontinuous Galerkin (RKDG) methods for the two-dimensional shallow water equations (SWE) are investigated. The \(p\)-adaptive algorithm that is implemented dynamically adjusts the order of the elements of an unstructured triangular grid based on a simple measure of the local flow properties of the numerical solution. Time discretization is accomplished using optimal strong-stability-preserving (SSP) RK methods. The methods are tested on two idealized problems of coastal ocean modeling interest with complex bathymetry – namely, the idealization of a continental shelf break and a coastal inlet. Numerical results indicate the stability, robustness, and accuracy of the algorithm, and it is shown that the use of dynamic \(p\)-adaptive grids offers savings in CPU time relative to grids with elements of a fixed order \(p\) that use either local \(h\)-refinement or global \(p\)-refinement to adequately resolve the solution while offering comparable accuracy.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
86A05 Hydrology, hydrography, oceanography

Software:

ADCIRC
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Full Text: DOI

References:

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