Solving the Laplace Equation in a Right-Angled Bicorn and Constructing Smooth Blending Functions for Conformal Overset Grids
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Solving the Laplace Equation in a Right-Angled Bicorn and Constructing Smooth Blending Functions for Conformal Overset Grids

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    A significant problem very often encountered when using a polyhedral grid framework for global numerical weather simulation is the ‘imprinting’ of spurious computational noise that occurs in the pattern of the edges and vertices of the chosen polyhedral geometry. Smoothing the grid across edges only partially mitigates the problem since the singularities still persist at the vertices. A more complete solution of this difficulty can be obtained by employing a composite overset grid configuration where two or more large grid domains overlap to cover the globe without singularities or regions of strong grid curvature. The redundancy of solutions in the overlap regions is then resolved by interpolations and progressive blending. It is preferable, for numerical reasons, to choose a grid that is orthogonal, especially if it possesses the additional desirable attribute of being conformal, provided this can be done in such a way that the resolution remains approximately uniform. A special class of these overset grids employs the idea of a two-sheeted Riemann surface, which enables the overlap regions of a conformal grid to be restricted to small two-cusp ‘bicorn’ areas very close to the polyhedron’s original vertices. At the cusps of each bicorn are a pair of ‘branch points’ of the mapping between the space of the grid and the geographical domain. While being the only singular points in the used part of the computational grid, these branch points still preserve continuity of several derivatives of the mapping and therefore behave as if they are effectively nonexistent as far as the model numerics are concerned. This note describes a general method of constructing the blending function for reconciling the two solutions in each bicorn based upon the principle of solving a Laplace equation for a ‘potential’ there (which is a valid solution to the Laplace equation in both sheets’ conformal coordinates) and deriving the weights as incomplete beta functions of that potential, in order to achieve a high degree of smoothness on the bounding edges of the bicorn.
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