A Discontinuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates

Abstract

A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations.The discontinuous Galerkin method consists of an integral formulation using a Rusanov numerical flux. A strong stability-preserving third order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, exept in a preprocessing step.The validation of the atmospheric model has been done considering steady-state and unsteady analytical solutions of the nonlinear shallow water equations. Experimental convergence was observed and the order of convergence k + 1 was achieved. Furthermore, the article presents a numerical experiment, for which the third order time-integration method limits the model error. Thus, the time step ∆t is restricted by both, the CFL-condition and accuracy demands. As a second step of validation, the model could reproduce a known barotropic instability caused by a small initial perturbation of a geostrophic balanced jet stream. Conservation of mass was shown up to machine precision and energy conservation converges with decreasing grid resolution and increasing polynomial order k.

IMEX HDG-DG: A coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems

“…Shallow water equations on a sphere. In this paper, we are also interested in the shallow water equations on the Earth surface, and for that reason, we consider the two-dimensional shallow water equations on the sphere with the Earth radius a = 6.371 × 10 6 m. We adopt the Lagrange multiplier approach [13,18,22,28], i.e., we embed the two-dimensional flow on the spherical manifold into the threedimensional space R 3 . The shallow water equation (3.2) on the spherical manifold can be cast into the following PDE in R 3 ∂q ∂t + ∇ · F = s in Ω, (3.6) where Ω is still the original surface of the sphere but now is considered a subset of R 3 , q := (φ, U) T := (φ, U, V, W ) T are the conservative variables, and…”

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

“…The tasks associated with flooding research of river valleys or interfluves [1] are stood out among the hydrological problems. In the framework of the SWM, the important results have also been obtained for the atmospheric phenomena, meteorological forecasts, and global climate models [8,9].…”

“…Among the numerous numerical methods solving shallow water equations (SWEs), the following methods should be mentioned: the discontinuous Galerkin method based on triangulation [8], the weighted surface-depth gradient method for the MUSCL scheme [18], and the modified finite difference method [20]. The so-called constrained interpolation profile/multimoment finite-volume method utilizing the shallow water approximation is developed to simulate geophysical currents on a rotating planet in spherical coordinate system ( [9] and see the references there).…”

Runge-Kutta Discontinuous Local Evolution Galerkin Methods for the Shallow Water Equations on the Cubed-Sphere Grid

“…The cubed-sphere grid is quasi-uniform and easily generated by dividing the sphere into six identical regions with the aid of projection of the sides of a circumscribed cube onto a spherical surface and choosing the coordinate lines on each region to be arcs of great circles. The mainly existing numerical methods for the SWEs on the sphere are as follows: finite-difference [2,39,46,47], finitevolume [21,24,51], multi-moment finite volume [6,7,22,23], spectral transform [16], spectral element [12,43,45], and discontinuous Galerkin (DG) methods [11,13,19,34,35] etc. Most of them are built on the one-dimensional exact or approximate Riemann solver.…”

“…19. Example 4.7: Relative vorticities at t = 6 days obtained by using by P 3 -based RKLEG method, P 3 -based RKDG method with Godunov's flux and FVLEG method with WENO5 reconstruction (from top to bottom) with N = 32.…”