Multiresolution analysis & simulation using spectral elements (MASUSE) Aim´ e Fournier October 7, 2006 Turbulence Numerics Team – 2006 NAR in collaboration with Duane Rosenberg & Annick Pouquet Sponsored by NSF (NCAR’s Programs) 1 Historical and scientific context From a mathematical point of view the fundamental challenge of computational fluid dynamics arises from nonlinear terms in the governing dynamical equations, of which the prototype is the flow velocity u and its advection (u ·  ∇)u. It is commonly known that the most accurate and efficient method to compute the gradient (  ∇) part is the spectral method, in which u(x) is approximated at each point x by a series in coefficients u  k of a set of smooth global basis functions such as {e i  k·x }  k∈K ; however the most efficient method (especially in massive parallel computation) for the multiplication part (·) is by a set {u(x   )}  ∈J of values corresponding to localized basis functions such as the finite elements e.g., the continuous function φ  ı (x) which equals δ  ı,  at x = x   and has uniform gradient inside certain disjoint simplexes that have vertices at the x   and cover the domain D. Unless the D geometry is trivial there is no known fast transform between the u  k and u(x   ), so some compromise, intermediate representation would be very desirable. One that combines the efficiency and geometric flexibility of finite elements with the accuracy of spectral methods is the spectral-element method [SEM, 16]. SEM was introduced to NCAR with application to both shallow-water [22] and 3D hydrostatic [9, 21, 23] dynamics on the sphere. The challenge of terms like (u ·  ∇)u is that they can lead to strongly localized multiscale features such as fronts, plumes and vortices, whose accurate description and prediction in regimes of realis- tically high Reynolds number (R, typical ratio of inertial to viscous force) exceeds the capability of today’s supercomputers if uniform, static meshes are used; this motivates the development of SEM with dynamically adaptive mesh refinement (AMR). Dynamic AMR-SEMs have been formulated for 1D [14] and 2D spherical [20] and planar [7, 11, 15] fluid-dynamics computations, in particular, GASpAR 1 [18]. Fournier [7] helped build connections between the methods of dynamic AMR-SEM and wavelets for PDEs; where the history of the latter spans 100s of reports for at least 17 years 2 [e.g., 1, 2, 17]. The methods described in the present report may be broadly labeled multiresolution analysis & simulation using spectral elements, or MASUSE. Multiresolution analysis generalizes Fourier analysis and is related to wavelet analysis, where the latter has been applied by NCAR scientists regularly for at least 12 years (at least 33 NCAR citations from [10] to [13]). 1 “GASpAR development” http://www.cisl.ucar.edu/nar/2006/catalog/Rosenberg1.pdf 2 Web of Science 2006/10/7 returns 352 documents with topic wavelet* and ("differential eq*" or pde*) 1