Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals
A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinearcoordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is use...
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| Language: | English |
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Yaroslavl State University
2015-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/283 |
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| author | E. V. Vorozhtsov V. P. Shapeev |
| author_facet | E. V. Vorozhtsov V. P. Shapeev |
| author_sort | E. V. Vorozhtsov |
| collection | DOAJ |
| description | A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinearcoordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniformity in the plane of the original polar coordinates is ensured with the aid of functions which control the grid stretching and entering the formulas of the passage from polar coordinates to the curvilinear ones. The method was verified on two test problems having exact analytic solutions. The examples of numerical computations show that if the radial coordinate axis origin lies outside the computational region, the proposed method has the second order of accuracy. If the computational region contains the singularity, the application of a non-uniform grid along the radial coordinate enables an increase in the numerical solution accuracy by factors from 1.7 to 5 in comparison with the uniform grid case at the same number of grid nodes. |
| format | Article |
| id | doaj-art-4849f497cd22483bbfde86ebc2479784 |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2015-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-4849f497cd22483bbfde86ebc24797842025-08-20T03:01:13ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172015-10-0122564866410.18255/1818-1015-2015-5-648-664264Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least ResidualsE. V. Vorozhtsov0V. P. Shapeev1Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Institutskaya str., 4/1, Novosibirsk, 630090, RussiaKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Institutskaya str., 4/1, Novosibirsk, 630090, Russia Novosibirsk National Research University, Pirogov str., 2, Novosibirsk, 630090, RussiaA version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinearcoordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniformity in the plane of the original polar coordinates is ensured with the aid of functions which control the grid stretching and entering the formulas of the passage from polar coordinates to the curvilinear ones. The method was verified on two test problems having exact analytic solutions. The examples of numerical computations show that if the radial coordinate axis origin lies outside the computational region, the proposed method has the second order of accuracy. If the computational region contains the singularity, the application of a non-uniform grid along the radial coordinate enables an increase in the numerical solution accuracy by factors from 1.7 to 5 in comparison with the uniform grid case at the same number of grid nodes.https://www.mais-journal.ru/jour/article/view/283poisson equationpolar coordinatesthe method of collocations and least residuals |
| spellingShingle | E. V. Vorozhtsov V. P. Shapeev Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals Моделирование и анализ информационных систем poisson equation polar coordinates the method of collocations and least residuals |
| title | Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals |
| title_full | Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals |
| title_fullStr | Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals |
| title_full_unstemmed | Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals |
| title_short | Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals |
| title_sort | numerical solution of the poisson equation in polar coordinates by the method of collocations and least residuals |
| topic | poisson equation polar coordinates the method of collocations and least residuals |
| url | https://www.mais-journal.ru/jour/article/view/283 |
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