A Discontinuous Galerkin Global Shallow Water Model

Abstract

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax-Friedrichs scheme. A third-order total variation diminishing Runge-Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

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

“…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 scalable high-order discontinuous Galerkin method for global atmospheric modeling*

“…Parallelism is effected through a hybrid MPI/OpenMP design and domain decomposition through the space-filling curve approach described in [3]. Our work extends earlier efforts [3,[10][11][12] in several important ways: first, we develop a scalable conservative 3-D DG-based dynamical core based on the hydrostatic primitive equations; second, we employ the vertical Lagrangian coordinate approach, developed by Starr [15] and later generalized by Lin [8]; and finally, we apply the 1-D cell-integrated semi-Lagrangian method [9] to preserve conservative remapping.…”

“…Toward this effort, the 2-D DG shallow water model in the HOMME framework [10,11] has been recently extended to 3-D DG baroclinic models [12]. Main features of DG baroclinic model are the vertical discretization and the prognostic equations which are based on hyperbolic conservation laws where as the prognostic variables are pressure thickness δp, covariant wind vectors (u 1 , u 2 ), potential temperature Θ, and moisture q.…”

“…The hydrostatic primitive equations in curvilinear coordinates employ the cubed-sphere geometry followed by [10,11]. A sphere is decomposed into '6 identical regions' by an equiangular central projection of the faces of an inscribed cube as displayed in Figure 1.…”

“…where the jump discontinuity at an element boundary requires the solution of a Riemann problem where the flux term F (U h ) · n can be approximated by a Lax-Friedrichs numerical flux as shown in [11]. The resulting DG discretization leads to following ordinary differential equation (ODE):…”