Remarks on Generation of the Orthogonal Structured Grids
Authors: Martynenko S.I. | Published: 19.02.2019 |
Published in issue: #1(82)/2019 | |
DOI: 10.18698/1812-3368-2019-1-16-26 | |
Category: Mathematics and Mechanics | Chapter: Computational Mathematics | |
Keywords: orthogonal grids, mathematical modeling, boundary value problems |
Grid generation techniques have contributed significantly toward the application of mathematical modeling in large-scale engineering problems. The structured grids have the advantage that very robust and parallel computational algorithms have been proposed for solving (initial-)boundary value problems. Orthogonal grids make it possible to simplify an approximation of the differential equations and to increase computation accuracy. Opportunity of the orthogonal structured grid generation for solving two- and three-dimensional (initial-)boundary value problems is analyzed in the article in assumption that isolines or isosurfaces of d (=2,3) functions form this grid. Condition of the isolines/isosurfaces orthogonality is used for formulation of the boundary value problems, the solutions of which will be form the orthogonal grid. A differential substitution is proposed to formulate the boundary value problems directly from the orthogonality condition of the grid. The substitution leads to the general partial differrential equations with undetermined coefficients. In the two-dimensional case, it is shown that the orthogonal grid generation is equivalent to the solution of partial differential equations of either elliptic or hyperbolic type. In three-dimensional domains, an orthogonal grid can be generated only in special cases. The obtained results are useful for mathematical modeling of the complex physicochemical processes in the technical devices
The work was supported by the RNF under the agreement no. 15-11-30012 of 07/08/2015 on the topic: "Supercomputer simulation of physical and chemical processes in the high-speed direct-flow propulsion jet engine of the hypersonic aircraft on solid fuels"
References
[1] Trottenberg U., Oosterlee C.W., Schüller A. Multigrid. Academic Press, 2001.
[2] Marchuk G.I. Metody vychislitelnoy matematiki [Computational mathematics methods]. Moscow, Nauka Publ., 1989.
[3] Hageman L.A., Young D.M. Applied iterative methods. Academic Press, 1981.
[4] Fedorenko R.P. A relaxation method for solving elliptic difference equations. USSR Comput. Math. & Math. Phys., 1962, vol. 1, iss. 4, pp. 1092–1096. DOI: 10.1016/0041-5553(62)90031-9
[5] Martynenko S.I. Mnogosetochnaya tekhnologiya: teoriya i prilozheniya [Multigrid technology: theory and applications]. Moscow, Fizmatlit Publ., 2015.
[6] Martynenko S.I. Robust multigrid technique for Black-Box software. DeGruyter, Berlin, 2017.
[7] Martynenko S.I. Robust multigrid technique for solving partial differential equations on structured grids. Vychislitelnye metody i programmirovanie [Numerical Methods and Programming], 2000, vol. 1, pp. 82--103 (in Russ.).
[8] Martynenko S.I., Volokhov V.M., Yanovskiy L.S. Parallel geometric multigrid. Int. J. Comp. Sci. Math., 2016, vol. 7, no. 4, pp. 293–300. DOI: 10.1504/IJCSM.2016.078741
[9] Thompson J.F., Soni B.K., Weatherill N.P. Handbook of grid generation. CRC Press, 1998.
[10] Liseikin V.D. Grid generation methods. Springer, 1999. Liseykin V.D., Likhanova Yu.V., Shokin Yu.I. Raznostnye setki i koordinatnye preobrazovaniya dlya chislennogo resheniya singulyarno vozmushchennykh zadach [Difference grids and coordinate transforms for numerical solution of singularly perturbed problems]. Novosibirsk, Nauka Publ., 2007.