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Generalized fibonacci grids; a new class of structured, smoothly adaptive multi-dimensional computational lattices
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    New class of structured, smoothly adaptive multi-dimensional computational lattices
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    This paper describes ways of extending the spatially-homogeneous two-dimensional computational 'Fibonacci grid' of Swinbank and Purser, and Hannay and Nye, to related grids exhibiting more general patterns of variable resolution and to higher dimensions. In principle, such a grid allows a single unified global computational framework to contain several independent subregions of enhanced resolution, blending smoothly with the rest of the grid where a uniform coarser resolution is maintained. Remarkably, it is able to do this without the geometry of the immediate neighborhood of any grid point (except near its two 'polar' singularities) exhibiting departures larger than a small fixed measure of deformation from a square, or cubic, lattice configuration. For meteorological data assimilation, such a grid has the advantage that the whole global domain's data can be assimilated in a single procedure that automatically accounts for the need to provide higher resolution in locations of special meteorological interest (cyclones, fronts, etc.). It may prove possible to use the new class of grids as the computational frameworks of numerical weather prediction models, in which case, the same advantages of unification carry over into this activity also. Associated with the higher-dimensional generalizations of these Fibonacci computational lattices we find generalizations, or analogues, of the Fibonacci and Lucas numbers themselves, but these new numbers occupy regular arrays having a dimensionality one less than the dimensionality of the space in which the Fibonacci lattices reside. It is shown that these 'Quasi-Fibonacci' numbers assume roles with respect to the generalized Fibonacci lattices exactly analogous to the roles played by the standard Fibonacci numbers in relation to the two-dimensional Fibonacci lattices, where they determine which generalized lines of the computational lattice can ever become suitable lines along which it is feasible to apply standard numerical operations of finite differencing, integration, filtering and interpolation.
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