Aime' Fournier, Gregory Beylkin, Vani Cheruvu

Multiresolution adaptive space refinement in geophysical fluid dynamics simulation

To efficiently and accurately numerically model highly nonlinear and intermittent geophysical fluid dynamics (GFD) is challenging. Classically, the finite-difference method (FDM) is most efficient, but the spectral method (SM) is most accurate. A method combining the best of FDM and SM would be especially appropriate for multiscale processes. The Spectral Element Method (SEM) is a candidate.

We present a recent generalization of SEM: Multiresolution Adaptive SEM Refinement (MASER). SEM decomposes the domain into finite elements, and is as efficient as FDM. Within each element, Gauss quadrature achieves SM accuracy. MASER enables SEM to adaptively refine resolution to dynamically emerging structures, while keeping high accuracy and efficiency. Not just multiscale, it is a multiresolution method: it represents functions and operators using bases covering a wide scale range, generated by scale transformations. From the 1-scale 1D, 1-element basis function u_i(x), one generates the multiresolution ND, K^N-element basis

u_{i,j,...k;p,q,...r;L}(x,y,...z) =

K^{N/2} u_i(Kx-p) u_j(Ky-q) ... u_k(Kz-r),

where K=2^L, (p,q,...r) indexes location and L indexes scale (wavenumber band K). Any basis at scale L spawns ``child'' bases at scale L+1, and adaptivity involves ``pruning the family tree'' (as in image compression). In a sense, it is Richardson's ``big whorls have little whorls'' idea cast as a rigorous numerical method. It is also more than adaptive mesh refinement --it is adaptive space refinement.

MASER is demonstrated on 2D tests that create a many strongly interacting scales, such as the 2D Burgers and Shallow Water Equations.