Original paper

Ocean Dynamics

, Volume 57, Issue 2, pp 109-121

First online:

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

  • Paul-Emile BernardAffiliated withCenter for Systems Engineering and Applied Mechanics (CESAME), Université Catholique de Louvain Email author 
  • , Nicolas ChevaugeonAffiliated withDépartement d’Architecture, d’Urbanisme de Génie Civil et Environnemental, Université Catholique de Louvain
  • , Vincent LegatAffiliated withCenter for Systems Engineering and Applied Mechanics (CESAME), Université Catholique de Louvain
  • , Eric DeleersnijderAffiliated withCenter for Systems Engineering and Applied Mechanics (CESAME), Université Catholique de LouvainInstitut d’Astronomie et de Géophysique G. Lemaître, Université Catholique de Louvain
  • , Jean-François RemacleAffiliated withCenter for Systems Engineering and Applied Mechanics (CESAME), Université Catholique de LouvainDépartement d’Architecture, d’Urbanisme de Génie Civil et Environnemental, Université Catholique de Louvain

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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-adaptivity Discontinuous Galerkin A posteriori error estimation