The Hybrid Discontinuous Galerkin method for elliptic problems and applications in vertical ocean-slice modeling (original) (raw)

Application of a coupled discontinuous–continuous Galerkin finite element shallow water model to coastal ocean dynamics

Ocean Modelling, 2005

A coupled discontinuous (DG) -continuous Galerkin (CG) shallow water model is compared to a continuous Galerkin generalized wave-continuity equation (GWCE) based model for the coastal ocean, whereby mass imbalance typical of GWCE-based solutions is eliminated using the coupled DG-CG model. Both realistic and idealized test problems for tide, wind, and wave-driven circulation form the basis of the study. Coupled DG-CG solutions retain the robustness of the well-documented solutions of GWCE-based models while more accurately capturing dynamics driven by small-scale, highly advective processes. The need for and sensitivity of the slope limiter in DG-based methods is shown and the utility of selective limiting demonstrated. Though the DG-based method is computationally more intensive this obstacle can be overcome through efficient parallelization. A model based on a coupled DG-CG formulation results in realistic and robust computations of coastal ocean dynamics.

A baroclinic discontinuous Galerkin finite element model for coastal flows

2012

Numerical modelling of coastal flows is a challenging topic due to complex topography of the coastal zone, rapid flow dynamics and large density variations. Such phenomena are best simulated with unstructured grid models due to their highly flexible spatial discretisation. This article presents a three-dimensional discontinuous Galerkin finite element marine model. Discontinuous Galerkin spatial discretisation is combined with an explicit mode splitting time integration scheme.

A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling

Ocean Modelling, 2009

We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic layer thickness and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water equations on an f -plane. The element pair has the property that all geostrophically balanced states which strongly satisfy the boundary conditions have discrete divergence equal to exactly zero and hence are exactly steady states of the discretised equations. This means that the finite element pair has excellent geostrophic balance properties. We also show that the element pair applied to the non-rotating linear shallow-water equations does not have any spurious small eigenvalues. We illustrate these properties using numerical tests and provide convergence calculations which show that the numerical solutions have errors which decay quadratically with element edge length for both velocity and layer thickness.

Continuous, discontinuous and coupled discontinuous–continuous Galerkin finite element methods for the shallow water equations

International Journal for Numerical Methods in Fluids, 2006

We consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) ÿnite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, ux-continuous on each element edge, and is suitable for both smooth and highly advective ows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous-continuous Galerkin method illustrates advantages and disadvantages in accuracy and e ciency. 64 C. DAWSON ET AL.

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

Computer Methods in Applied Mechanics and Engineering, 2002

We consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) ÿnite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, ux-continuous on each element edge, and is suitable for both smooth and highly advective ows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous-continuous Galerkin method illustrates advantages and disadvantages in accuracy and e ciency. 64 C. DAWSON ET AL.

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

Ocean Dynamics, 2007

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 pþ1 , 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.

A discontinuous Galerkin method for two-layer shallow water equations

Mathematics and Computers in Simulation, 2015

In this paper, we study a discontinuous Galerkin method to approximate solutions of the two-layer shallow water equations on non-flat topography. The layers can be formed in the shallow water model based on the vertical variation of water density which in general depends on the water temperature and salinity. For a water body with equal density the model reduces to the canonical single-layer shallow water equations. Thus, for a model with equal density on flat bottom, the method is equivalent to the discontinuous Galerkin method for conservation laws. The considered method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hpadaptivity. Several numerical results illustrate the performance of the method and confirm its capability to solve two-layer shallow water flows including tidal conditions on the water free-surface and bed frictions on the bottom topography.

A Discontinuous Galerkin Method for Three-Dimensional Shallow Water Equations

Journal of Scientific Computing, 2005

We describe the application of a local discontinuous Galerkin method to the numerical solution of the three-dimensional shallow water equations. The shallow water equations are used to model surface water flows where the hydrostatic pressure assumption is valid. The authors recently developed a DG method for the depth-integrated shallow water equations. The method described here is an extension of these ideas to non depth-integrated models. The method and its implementation are discussed, followed by numerical examples on several test problems.

Distributed Implementation and Verification of Hybridizable Discontinuous Galerkin Methods for Nonhydrostatic Ocean Processes

OCEANS 2018 MTS/IEEE Charleston, 2018

Nonhydrostatic, multiscale processes are an important part of our understanding of ocean dynamics. However, resolving these dynamics with traditional computational techniques can often be prohibitively expensive. We apply the hybridizable discontinuous Galerkin (HDG) finite element methodology to perform computationally efficient, high-order, nonhydrostatic ocean modeling by solving the Navier-Stokes equations with the Boussinesq approximation. In this work, we introduce a distributed implementation of our HDG projection method algorithm. We provide numerical experiments to verify our methodology using the method of manufactured solutions and provide preliminary benchmarking for our distributed implementation that highlight the advantages of the HDG methodology in the context of distributed computing. Lastly, we present simulations in which we capture nonhydrostatic internal waves that form as a result of tidal interactions with ocean topography. First, we consider the case of tidal...

A high-order triangular discontinuous Galerkin oceanic shallow water model

International Journal for Numerical Methods in Fluids, 2008

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.