Nodal High-Order Discontinuous GalerkinMethods for the Spherical ShallowWater Equations

We develop and evaluate 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 LegendreGauss-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 as well as the impact of over-integration, confirm the expected high-order accuracy and the potential for using such highly parallel formulations in numerical weather prediction.