Nodal High-Order Discontinuous Galerkin Methods for the Spherical Shallow Water Equations

Abstract

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. c 2002 Elsevier Science (USA)

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

“…The same experimental order of convergence is obtained by spectral element methods, see [11,18]. Further for fixed grid resolutions Dx, the errors decrease significantly for increasing k. These results are very close to the convergence studies in [16,30,15]. For this steady-state solution no limitation due to the third-order RK method is observable.…”

“…Numerous numerical methods have been proposed for next generation global atmospheric models including finite volumes [27,34], spectral elements [45,11,9,17], and DG [16,30,15] methods. We have selected the DG method for our model because it allows us to achieve high-order accuracy as in spectral elements while conserving all quantities both locally and globally as in finite volumes, see the review in [5].…”

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

“…Significant progress in the application of DG methods to the SWE has been achieved in the last few years [15,16,17,18,19,20,21,22,23,24,25]. However, two issues relevant in many applications, namely preserving steady-states at rest with variable bathymetry and properly handling flooding and drying, have not been addressed in previous work, with the exception of [17] where a moving mesh was used to deal with dry areas in a one-dimensional setting; the extension to two space dimensions does not seem to be straightforward.…”

A high-order conservative collocation scheme and its application to global shallow-water equations

“…Among many others are those reported in Giraldo et al (2002), Thomas and Loft (2005), Giraldo and Warburton (2005), Nair et al (2005a, b), Taylor and Fournier (2010) and Blaise and StCyr (2012). Two major advantages that make these models attractive are (1) they can reach the targeted numerical accuracy more quickly by increasing the number of degrees of freedom (DOFs) (or unknowns), and (2) they can be more computationally intensive with respect to the data communications in parallel processing (Dennis et al, 2012).…”