Abstract
In this work, the three-dimensional Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly, by extending the method of Hockney. The Poisson equation is approximated by second-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results.
Highlights
The three-dimensional Poisson’s equation in cylindrical coordinates r, z is given by U rr 1 r Ur 1 r2 U U zz f r, z (1)which is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics
1 r Ur which is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics
We develop a second-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [14] and extend the Hockney’s method [15] to solve the three dimensional Poisson’s equation on Cylindrical coordinates system
Summary
The three-dimensional Poisson’s equation in cylindrical coordinates r, , z is given by. The analytic solution for the three-dimensional Poisson’s equation in cylindrical coordinate system is much more complicated and tedious because of the complexity of the nature of the problems and their geometry, and the availability of appropriate methods. To solve Poisson’s equation in polar and cylindrical coordinates geometry, different approaches and numerical methods using finite difference approximation have been developed. We develop a second-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [14] and extend the Hockney’s method [15] to solve the three dimensional Poisson’s equation on Cylindrical coordinates system
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