Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations

By F. Giraldo, Jan S. Hesthaven, Tim Warburton. 2002

279 citations recorded by the source corpus.

We present a high-order discontinuous Galerkin method for the solution of the shallow water equations on the sphere. To overcome well-known problems with polar singularities, we consider the shallow water equations in Cartesian coordinates, augmented with a Lagrange multiplier to ensure that fluid particles are constrained to the spherical surface. The global solutions are represented by a collection of curvilinear quadrilaterals from an icosahedral grid. On each of these elements the local solutions are assumed to be well approximated by a high-order nodal Lagrange polynomial, constructed from a tensor-product of the Legendre-Gauss-Lobatto points, which also supplies a high-order quadrature. The shallow water equations are satisfied in a local discontinuous element fashion with solution continuity being enforced weakly. The numerical experiments, involving a comparison of weak and strong conservation forms and the impact of over-integration and filtering, confirm the expected high-order accuracy and the potential for using such highly parallel formulations in numerical weather prediction.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/600f30f9485a9b66d6af1c322003a078f8395676

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Numerical Analysis

Citations in the corpus

The Evolution of Dynamical Cores for Global Atmospheric Models(125th Anniversary Issue of the Meteorological Society of Japan) (2007, methodology): “They used a modal expansion in comparison to the nodal expansion of Giraldo et al. (2002).”
High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model (2009, background): “, [15, 29, 30]) but they do not exist for the triangle.”
A Discontinuous Galerkin Transport Scheme on the Cubed Sphere (2005, methodology): “Giraldo et al. (2002) have used the 3D Cartesian equations to solve a 2D problem on the sphere.”
A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying (2008, methodology): “Significant progress in the application of DG methods to the SWE has been achieved in the last few years for various applications, like dam break problems and flows in channels [18, 19, 20, 21, 22, 23, 24], harbor wave disturbances and tidal flows [21, 25, 26, 27], flooding and drying [22], or geoph”
A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations (2004, methodology): “[16].”
A Survey of High Order Schemes for the Shallow Water Equations (2014, background): “Several advantages of the DGmethod, including its high order accuracy, high parallel efficiency, flexibility for hp-adaptivity and arbitrary geometry and meshes, make it suited for the SWEs, see [16, 19, 23, 38, 51].”
Horizontal grids for global weather and climate prediction models: a review (2012, methodology): “Decomposition of a triangle into quadrilaterals is used in Giraldo (1998, 2001), Giraldo et al. (2002) and Giraldo and Rosmond (2004) to define quadrilateral spectral elements on the twenty triangular faces of an icosahedron; Figure 3 of Giraldo and Rosmond (2004) shows examples of this with two…”
Dynamic p-adaptive Runge–Kutta discontinuous Galerkin methods for the shallow water equations (2009, background): “…all hp finite element methods for the twodimensional shallow water equations (SWE) that have appeared in the literature have used this approach, i.e. they have utilized local h-refinement only, while using a globally defined p that is chosen prior to the computations (see, e.g., [6] [7] [9] [11]).”
A Discontinuous Galerkin Global Shallow Water Model (2005, methodology): “An alternative approach is a nodal basis where the basis set is constructed using Lagrange–Legendre polynomials with roots at Gauss– Lobatto quadrature points (Giraldo et al. 2002).”
A high‐order triangular discontinuous Galerkin oceanic shallow water model (2008, methodology): “[10] first used the DG method for the shallow water equations on the sphere, which was later followed by the construction of such models on triangular domains (see [11]); however, our work has focused on the nodal space approximations (Lagrange polynomials) rather than the more commonly used modal s”
On the accuracy of high-order discretizations for underresolved turbulence simulations (2013, unknown): “DGSEM is used in a wide range of applications such as compressible flows [5], [6], [33], electromagnetics and optics [28], [29], [11], [12], aeroacoustics [34], [36], [37], [9], meteorology [18], [19], [35], and geophysics [13], [14].”
hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow (2006, methodology): “In [15,17] highorder global DG SWE models, which solve the equations on the sphere using curvilinear quadrilateral elements, were developed and applied to a number of tests cases for problems of a global scale (e.”
A New Class of High-Order Energy Stable Flux Reconstruction Schemes (2011, background): “Such schemes have become known as nodal DG methods, an exposition of which can be found in the recent textbook by Hesthaven and Warburton [4], as well as in various articles by the same authors [5, 6].”
High-order h-adaptive discontinuous Galerkin methods for ocean modelling (2007, background): “Dynamic mesh adaptation strategies following those structures represent a great potential in the field of ocean modelling (e.g. Behrens 1998; Heinze and Hense 2002; Nair et al. 2005; Giraldo et al. 2002).”
An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods (2006, methodology): “[5] use a fast quadrature free DGM for solving the spherical SWEs.”