Multi-scale geophysical modeling using the spectral element method

Climate modeling encompasses an enormous range of spatial and temporal scales. One of its great remaining challenges is bridging the scale gaps between global climate processes, basinscale and regional impacts, and smaller-scale ecosystem dynamics. Improving and accessing enhanced computational resources will help bridge this gap. Nonetheless, enhanced computer power alone is insufficient without parallel improvements in numerical algorithms.1 The geophysical modeling community is exploring several approaches to address the issues of multiscale simulations in geometrically complex regions, including finite-element and finite-volume methods. These methods are especially attractive because of the geometric flexibility inherent in their unstructured computational grids. Modelers can adjust the shape, size, orientation, and connectivity of the cells forming this grid to fit the geometric and dynamical constraints of the problem at hand. Particularly, they can use a single grid with variable cell sizes to address the various requirements of multiscale simulations. One such approach is the spectral element method. This method is an h-p type finite-element method that combines the geometrical flexibility of traditional, commonly low-order, finiteelement methods and the high-order accuracy normally associated with spectral methods. (For a history of the spectral element method, see the related sidebar.) In a spectral element solution, the computational domain is divided into a finite number of cells, called elements, where the solution is interpolated with a high-degree polynomial. The spectral element method offers several attractive properties for geophysical simulations: