Nodal integral methods for the numerical solution of the shallow-water equations

A series of new coarse-mesh numerical methods for the solution of the two-dimensional shallow-water equations is developed, first for the beta-plane model and then on the surface of a sphere. As with other methods of this type (so-called "nodal" methods), the partial differential equations are transverse averaged over each computational element or "node," and the ordinary differential equations resulting from this procedure are solved analytically within each element, with the solutions being written in terms of the dependent variables averaged along the element boundaries. The remaining parts of the equations, which are collected in source-like terms, are approximated via low-order expansions to complete the discretization process. The resulting set of nonlinear algebraic equations is solved iteratively using a Newton-Krylov method.; Three new methods are developed for the shallow-water model in the beta-plane with a second-order, linear viscosity term. The first method is developed for the steady-state shallow-water model. This method is extended to the time-dependent equations by approximating the temporal derivatives with a simple finite-difference scheme, thereby resulting in a hybrid method combining features from both the nodal integral and finite-difference approaches. Finally, a method is developed for the time-dependent equations by applying the nodal integral approach to both the spatial and temporal terms.; Each new method in the beta-plane is applied to several test problems, the analytical solutions of which are available, and the numerical results are compared to these analytical solutions. The third method also is applied to other elementary flow problems for which analytical solutions are not available. The numerical results at several resolutions are compared to demonstrate that the method provides accurate results on coarse meshes.; Next, a systematic treatment of the transverse averaging procedure for general orthogonal curvilinear coordinates is developed. This new technique is used to develop a new method for the continuity equation in spherical coordinates which is tested by advecting a test pattern around the surface of a sphere.; Finally, a different, new nodal integral method that conserves both mass and absolute vorticity is developed for the inviscid shallow-water equations in spherical coordinates. This method is tested using several selected benchmark problems for this system of equations. The accuracy of the method is discussed and compared to other methods for the shallow-water equations on the sphere.