Geodesic
On some method of construction of elliptical grids assigned to explicit analytical expressions
Matematičeskoe modelirovanie, Tome 15 (2003) no. 6, pp. 101-106.

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Now algebraic approaches to construction of grids are rapidly developing (B-spline, transfinite interpolation etc.) [1,2]. Their advantage appears in the possibility of fast construction of grids. Traditional approaches based on a numerical solution of differential equations give, as a rule, best grids, but demand much more time for their construction. Especially it is right in calculation of flows in moving boundaries. In [4], the new method of grid construction based on an analytical solution of some system of elliptic differential equations is suggested. It combines a speed of the algebraic approach with quality of the elliptical approach. In suggested method, the functions which map computational space to physical one (mapping-functions, MF), are obtained in exact form. In present work, an approach [4] is generalized. MF are constructed here directly, avoiding solutions of any differential equations. In spite of it, these grids have all positive features of elliptic grids. Obtained results are illustrated with numerous examples. It allows to receive analytical expressions for metric coefficients and, thereby, completely eliminate errors of their numerical calculation. The method of control of some relevant grid properties is also suggested. The idea of quality control of the obtained grid is based on application of analytical solutions of a boundary value problem for the class of higher order equations. As MF remain analytical, method does not increase grid construction time. The offered grids have some advantage in comparison to conformal ones, where the MF have no sufficient functional arbitrariness. The obtained results are illustrated by examples. 
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     author = {E. I. Sokolov and N. B. Fedosenko},
     title = {On some method of construction of elliptical grids assigned to explicit analytical expressions},
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E. I. Sokolov; N. B. Fedosenko. On some method of construction of elliptical grids assigned to explicit analytical expressions. Matematičeskoe modelirovanie, Tome 15 (2003) no. 6, pp. 101-106. http://geodesic.mathdoc.fr/item/MM_2003_15_6_a16/

[1] Thompson J. F., Warsi Z. U. A., Mastin C. W., Numerical grid generation. Foundations and applications, North Holland, 1985 | MR | Zbl

[2] Bramkamp F., Ballmann J., Muller S., “Development of a flow solver employing local adaptation bazed on multiscale analysis on $B$-spline grids”, Proceedings of 8th Annual Conf. of the CFD Society of Canada (June, 11 to 13, 2000, Montreal)

[3] Brakhage S., Muller K., “Algebraic-hyperbolic grid generation with precise control of intersection of angles”, Int. J. Numer. Meth. Fluids, 33:1 (2000), 89–123 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[4] Fedosenko N., Sokolov E., “Application of exact solutions of some elliptic equations for generation of two- and multidimentional analytical grids”, Proceedings of ALGORITMY 2002 Conference of Scientific Computing, 253–259