A Spectral Element Semi-lagrangian Method for the Shallow Water Equations on Unstructured Grids

Abstract. The purpose of this paper is twofold: to give a brief yet comprehensive description of the spectral element and semi-Lagrangian methods, and to introduce a new method arrived at by fusing both of these impressive methods. The practical aspects of both methods are described in detail by their implementation on the 2D shallow water equations. These equations have been used customarily to develop new numerical methods for weather prediction models because they exhibit the same wave behavior as the more complex 3D atmosphere and ocean equations. The spectral element method is essentially a higher order finite element method that exhibits spectral convergence provided that the solution is a smooth function. The semi-Lagrangian method traces the characteristic curves of the solution and, consequently, is very well suited for resolving the non-linearities introduced by the advection operator of the fluid dynamics equations. By using the basis functions of the spectral element method as the interpolation functions for the semi-Lagrangian method, one can create a truly local method that can be used in conjunction with unstructured/adaptive grid generation and can be ported quite naturally to parallel computer architectures. Results for the fully non-linear 2D shallow water equations are presented thus showing the benefits of this new scheme.