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

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.