Parallel Spectral Element Atmospheric Model

Abstract

This chapter presents a semi-implicit formulation of the parallel spectral element atmospheric model (SEAM). Spectral element methods are h-p type finite element methods that combine the geometric flexibility of finite elements with the exponential convergence of pseudospectral methods. Spectral elements combine the accuracy of conventional spectral methods and the geometric flexibility of finite element methods. In a spectral element discretization, the computational domain is divided into rectangular elements, within which variables are approximated by a polynomial expansion of high degree. The discrete equations are derived using Gauss–Lobatto–Legendre quadrature together with the Lagrangian interpolants on the collocation grid (Legendre cardinal functions) as basis functions, resulting in diagonal mass matrices. There are several practical advantages to using a relatively new, high-accuracy numerical method such as the spectral element method (SEM) over current methods. Spectral elements have desirable boundary-exchange communication patterns that are reminiscent of finite difference models. Fast computation and linear scaling curves can ultimately result in a useful kernel for climate simulation only if the amount of model time computed per unit wall clock time is sufficiently large. An efficient semi-implicit solver is essential if this all important performance metric is to be substantially increased.