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

A global 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, 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 twodimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge–Kuttamethod is applied for the timediscretization. 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, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L.Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallowwater equations in spherical geometry, J. Comput. Phys. 102 (1992) 211–224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of OðDxkþ1Þ was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integrationmethod limits themodel error. Thus, the time stepDt is restrictedbyboth theCFL-conditionandaccuracy demands. Conservation ofmasswas shownup tomachine precision andenergy conservation converges for both increasing grid resolution and increasing polynomial order k. 2008 Elsevier Inc. All rights reserved.