Abstract
In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-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,θ, z ) is given by U rr + 1 r Ur 1 r2 Uθθ + U zz = f (r,θ, z) (1)has a wide range of application in engineering and science fields.How to cite this paper: Shiferaw, A. and Mittal, R.C. (2014) High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate
We develop a fourth-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [9] [10] and extend the Hockney’s method [9] [11] to solve the three dimensional Poisson’s equation on Cylindrical coordinates system
All these set of Equations (20a)-(20c) are tri-diagonal ones and we solve for vijk by using Thomas algorithm
Summary
The three-dimensional Poisson’s equation in cylindrical coordinates (r,θ , z ) is given by. (2014) High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate. For the numerical solution of the three dimensional Poisson’s equation in cylindrical coordinates system, several attempts have been made in particular for physical problems that are related directly or indirectly to this equation. The need to obtain the best solution for the three dimensional Poisson’s equation in cylindrical coordinates system is still in progress. We develop a fourth-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [9] [10] and extend the Hockney’s method [9] [11] to solve the three dimensional Poisson’s equation on Cylindrical coordinates system
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