Spectral Element Method Part 1 : Numerical

SPECTRAL ELEMENT METHOD PART 1: NUMERICAL ALGORITHMS Aimé Fournier, Joseph Tribbia National Center for Atmospheri Resear h Boulder CO 80307-3000 USA fournier u ar.edu Mark A. Taylor Albuquerque NM 87108 USA www.s d.u ar.edu/ ss/staff/taylor ABSTRACT It is well known that the fast and a urate solution of the partial di erential equations (PDEs) governing geophysi al uid dynami s presents a great hallenge. This is due to the omplexity of both the PDEs themselves, and the initial and boundary onditions. There are several pra ti al advantages to using a relatively new, high-a ura y numeri al method, the Spe tral Element Method (SEM), over urrently standard methods, to ompute su h solutions. Put su in tly, SEM ombines the a ura y of onventional spe tral methods and the geometri exibility of nite element methods. Figure 1 shows the gnomoni proje tion to the sphere from the ube tiled with 6 42 elements, yielding a quasi-uniform dis retization. This paper has several purposes. The advantages of SEM with regard to exibility, a ura y and e ient parallel implementation are explained, in order to motivate its adoption where appropriate. Disadvantages are also mentioned. We des ribe the Spe tral Element Atmospheri Model (SEAM), in luding su ient details that the readers may estimate the bene t of applying SEM to their own omputations. Another goal is to remedy the urrent la k of a des ription of SEM whi h is a essible to an audien e other than advan ed spe ialists in the numeri al solution of PDEs.INTRODUCTION We begin by explaining in detail the mathemati al elements of the spe tral element method (SEM), as des ribed by the ompanion paper by Thomas et al. [1℄ elsewhere in the present volume. These in lude the quadrature, oordinate maps and representation of operators. We also review the time-stepping method [2 4℄, noting its advantages and limitations ompared to the pseudospe tral ase. Improvements are addressed in ref. 1. In the Results se tion we show the advantages of lo al mesh re nement. A B C D E F G H I