Discontinuous finite-element methods for two- and three-dimensional marine flows

Numerical modeling is now, along with experiment and theory, part of the scientific method. Numerical models for marine flows exist for more than forty years. These models have evolved dramatically, with improved numerical methods, and much more accurate modeling of unresolved phenomena. However, most of the mainstream marine models still rely on the old numerical paradigm, based on finite difference methods and structured grids. The development of new numerical models, based on state-of-the art numerical methods on unstructured grids, is now an area of active research. Those unstructured meshes allow a faithful representation of the coastlines and the choice of the resolution following guidelines from the physics and not from the numerics. This thesis fits within this research. It is part of the development of SLIM (http://www.climate.be/SLIM), the Second-generation Louvain-la-neuve Ice-ocean Model. This model uses finite element methods on unstructured meshes made up of triangles for two-dimensional modeling, and triangular prisms for three-dimensional modeling. Numerical aspects are considered in this work. First, a comparison of several finite-element discretizations of the two-dimensional shallow water equations is performed. Second, to handle flows on the sphere, a novel algorithm is described, based on local coordinate systems, that allows a discretization free from singularities. Finally, a prototype three-dimensional baroclinic model is presented. The model features a spatial discretization built upon discontinuous finite elements and a time integration performed with an implicit mode splitting.(FSA 3) -- UCL, 201

A finite element method for solving the shallow water equations on the sphere

Within the framework of ocean general circulation modeling, the present paper describes an efficient way to discretize partial differential equations on curved surfaces by means of the finite element method on triangular meshes. Our approach benefits from the inherent flexibility of the finite element method. The key idea consists in a dialog between a local coordinate system defined for each element in which integration takes place, and a nodal coordinate system in which all local contributions related to a vectorial degree of freedom are assembled. Since each element of the mesh and each degree of freedom are treated in the same way, the so-called pole singularity issue is fully circumvented. Applied to the shallow water equations expressed in primitive variables, this new approach has been validated against the standard test set defined by [Williamson, D.L., Drake, J.B., Hack, J.J., Jakob, R., Swarztrauber, P.N., 1992. A standard test set for numerical approximations to the shallow water equations in spherical geometry. journal of Computational Physics 102, 211-224]. Optimal rates of convergence for the P-1(NC) - P-1 finite element pair are obtained, for both global and local quantities of interest. Finally, the approach can be extended to three-dimensional thin-layer flows in a straightforward manner. (C) 2008 Elsevier Ltd. All rights reserved

Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second-order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (p(1)(NC)-P-1(NC),) is presented, giving optimal rates of convergence in all test cases. P-1(NC)-P-1 and P-1(DG)-P-1 mixed formulations lack convergence for inviscid flows. P-1(DG)-P-2 pair is more expensive but provides accurate results for all benchmarks. P-1(DG)-P-1(DG) provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed. Copyright (C) 2009 John Wiley & Sons, Ltd

Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

This paper presents multirate explicit time-stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large-scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes : Part II: implicit/explicit time discretization

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other author

A flux-limiting wetting-drying method for finite-element shallow-water models, with application to the Scheldt Estuary

We present a flux-limiting wetting-drying approach for finite-element discretizations of the shallow-water equations using discontinuous linear elements for the elevation. The key ingredient of the method is the use of limiters for generalized nodal fluxes. This method is implemented into the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM), and is verified against standard test cases. The method is further applied to the wetting and drying of sand banks in the Scheldt Estuary, which is located in northern Belgium and the southern Netherlands. The results obtained for both the benchmarks and the realistic problem illustrate the accuracy of the method in describing the hydrodynamics in the vicinity of dry areas. In particular, the method strictly conserves mass, and there is no transport through dry areas. (C) 2009 Elsevier Ltd. All rights reserved

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes : Part I: space discretization

We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic threedimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward

A fully implicit wetting-drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary

Resolving the shoreline undulation due to tidal excursion is a crucial part of modelling water flow in estuaries and coastal areas. Nevertheless, maintaining positive water column depth and numerical stability has proved out to be a very difficult task that requires special attention. In this paper we propose a novel wetting–drying method in which the position of the sea bed is allowed to fluctuate in drying areas. The method is implemented in a Discontinuous Galerkin Finite Element Model (DG-FEM). Unlike most methods in the literature our method is compatible with fully implicit time-marching schemes, thus reducing the overall computational cost significantly. Moreover, global and local mass conservation is guaranteed which is crucial for long-term environmental applications. In addition consistency with tracer equation is also ensured. The performance of the proposed method is demonstrated with a set of test cases as well as a real-world application to the Scheldt Estuary. Due to the implicit time integration, the computational cost in the Scheldt application is reduced by two orders of magnitude. Although a DG-FEM implementation is presented here, the wetting–drying method is applicable to a wide variety of shallow water models

The vertical age profile in sea ice: Theory and numerical results

The sea ice age is an interesting diagnostic tool because it may provide a proxy for the sea ice thickness and is easier to infer from observations than the sea ice thickness. Remote sensing algorithms and modeling approaches proposed in the literature indicate significant methodological uncertainties, leading to different ice age values and physical interpretations. In this work, we focus on the vertical age distribution in sea ice. Based on the age theory developed for marine modeling, we propose a vertically-variable sea ice age definition which gives a measure of the time elapsed since the accretion of the ice particle under consideration. An analytical solution is derived from Stefan’s law for a horizontally homogeneous ice layer with a periodic ice thickness seasonal cycle. Two numerical methods to solve the age equation are proposed. In the first one, the domain is discretized adaptively in space thanks to Lagrangian particles in order to capture the age profile and its discontinuities. The second one focuses on the mean age of the ice using as few degrees of freedom as possible and is based on an Arbitrary Lagrangian–Eulerian (ALE) spatial discretization and the finite element method. We observe an excellent agreement between the Lagrangian particles and the analytical solution. The mean value and the standard deviation of the finite element solution agree with the analytical solution and a linear approximation is found to represent the age profile the better, the older the ice gets. Both methods are finally applied to a stand-alone thermodynamic sea ice model of the Arctic. Computing the vertically-averaged ice age reduces by a factor of about 2 the simulated ice age compared to the oldest particle of the ice columns. A high correlation is found between the ice thickness and the age of the oldest particle. However, whether or not this will remain valid once ice dynamics is included should be investigated. In addition, the present study, based on thermodynamics only, does not support a single age-thickness functional relationship