# High-order *h*-adaptive discontinuous Galerkin methods for ocean modelling

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DOI: 10.1007/s10236-006-0093-y

- Cite this article as:
- Bernard, P., Chevaugeon, N., Legat, V. et al. Ocean Dynamics (2007) 57: 109. doi:10.1007/s10236-006-0093-y

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## Abstract

In this paper, we present an *h*-adaptive discontinuous Galerkin formulation of the shallow water equations. For a discontinuous Galerkin scheme using polynomials up to order \(
p
\), the spatial error of discretization of the method can be shown to be of the order of \(
h^{{p + 1}}
\), where \(h\) is the mesh spacing. It can be shown by rigorous error analysis that the discontinuous Galerkin method discretization error can be related to the amplitude of the inter-element jumps. Therefore, we use the information contained in jumps to build error metrics and size field. Results are presented for ocean modelling problems. A first experiment shows that the theoretical convergence rate is reached with the discontinuous Galerkin high-order *h*-adaptive method applied to the Stommel wind-driven gyre. A second experiment shows the propagation of an anticyclonic eddy in the Gulf of Mexico.

### Keywords

Shallow water equations*H*-adaptivityDiscontinuous GalerkinA posteriori error estimation