Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals
https://doi.org/10.18255/1818-1015-2015-5-648-664
Abstract
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 curvilinear
coordinates 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.
About the Authors
E. V. Vorozhtsov
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences, Institutskaya str., 4/1, Novosibirsk, 630090, Russia
Russian Federation
Doctor of physical and mathematical sciences, professor
Russian Federation
Doctor of physical and mathematical sciences, professor
V. P. Shapeev
Khristianovich 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, Russia
Russian Federation
Doctor of physical and mathematical sciences, professor
Russian Federation
Doctor of physical and mathematical sciences, professor
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Review
For citations:
Vorozhtsov E.V., Shapeev V.P. Numerical Solution of the Poisson Equation in Polar Coordinates by the Method of Collocations and Least Residuals. Modeling and Analysis of Information Systems. 2015;22(5):648-664. (In Russ.) https://doi.org/10.18255/1818-1015-2015-5-648-664
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