High-Order Discontinuous Galerkin Schemes on General 2D Manifolds
and
Multiscale Mesh Generation on the Sphere

Part One
An innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the Discontinuous Galerkin Method. This new approach combines the advantages of the usual approaches described in the literature. The vectorial fields are expressed in a unit non-orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements, while the convective flux functions are written is the usual 3D Cartesian coordinate system. The number of vectorial components is therefore minimum and the tangency constraint is naturally ensured, while the method remains robust and general since not relying on a particular parametrization of the manifold. The Discontinuous Galerkin Method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition. The possible discontinuities of this basis are then taken into account in the Riemann solver on inter-element interfaces. The approach is validated on the sphere, using the shallow water equations for computing standard atmospheric benchmarks. In particular, the Williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error. The propagation of gravity waves is eventually computed on non-conventional irregular curved manifolds to illustrate the robustness and generality of the method.

Part Two
A method for generating computational meshes for applications in ocean modeling is presented. The method uses a standard engineering approach for describing the geometry of the domain that requires meshing. The underlying sphere is parametrized using stereographic coordinates. Then, coastlines are described with cubic splines drawn in the stereographic parametric space. The mesh generation algorithm builds the mesh in the parametric plane using available techniques. The method enables to import coastlines from different data sets, and consequently to build meshes of domains with highly variable length scales. The results include meshes together with numerical simulations of various kinds.