Algorithm for the numerical solution of the Poisson’s equation using the finite difference
Keywords:
algorithm, Poisson’s equation, differential equations in partial derivatives, numerical solution, finite differences
Abstract
The algorithm is designed from a method that results in an adaptation of the finite difference method for boundary value problems. In the procedure, the iterative Gauss-Seidel method is used to solve the linear system produced. Different network sizes in the axes are allowed in the structure. To visualize the execution of the algorithm, the numerical solution of a particular problem is included.
References
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Al-Jawary, M. & Hatif, S. A semi-analitical iterative method for solving differential algebraic equations, Ain Chams Engeneering Journal, (1989), doi: 10.1016/j.asej.2017. 07.004.
Beiro da Veiga, L., Lopez, L. & Vacca, G. Mimetic finite difference methods for Hamiltonian wave equations in 2D, Computer and Matematics with applications, (2017), doi: 10.1016/j.camwa.2017.05.022.
Cacasús Acevedo, A. Aplicación del método de diferencias finitas generalizadas a problemas de elasto-dinámica. (Tesis inédita de doctorado), E.T.S. de Ingenieros Industriales UNED, Madrid, España, 2011.
De la Fuente, J.L. Técnicas de cálculo para sistemas de ecuaciones, programación lineal y programación entera: códigos en FORTRAN y C con aplicaciones de sistemas de energía eléctrica., Reverté, Madrid, España, 1997.
Fernández, L. A. Introducción a las ecuaciones en derivadas parciales, Universidad de Cantambria, Cantambria, España, 2016.
Gosse, L. Dirichlet-to-neumann mapping and finite-differences for anisotorpic difusion, Computers and Fluids, 156 (2017), 58–65.
Quintanilla Murillo, J. Métodos numéricos en diferencias finitas para la resolución de ecuaciones difusivas fraccionarias. (Tesis inédita de doctorado), Universidad de Extremadura, Badajoz, España, 2016.
Varga, R.S. Matrix iterative analysis., Prentice Hall, Englewood Cliffs, New Jersey, United State of America, 1962.
Wang, H., Liang, H., & Chai, Z. Finite-diference lattice Boltzmann model for nonlinear convection-diffusion equations, Applied Mathematics and Computation, 309 (2017), 334–349, doi: 10.1016/j.amc. 2017.04.015.
Xue, G. & Zhang, L. A new finite difference shemes for generalized Rosenau-Burgers equation, Applied Mathematics and Computation, 222(2013), 490–496, doi: 10.1016/j.amc.2013.07.052.
Zapata, M. U. & Itzá, R. High-order implicit finite difference shemes for the two dimensional Poisson equation, Applied Mathematics and Computation, 309(2017), 222–244, doi: 10.1016/j.1mc.2017.04.006.
Al-Jawary, M. & Hatif, S. A semi-analitical iterative method for solving differential algebraic equations, Ain Chams Engeneering Journal, (1989), doi: 10.1016/j.asej.2017. 07.004.
Beiro da Veiga, L., Lopez, L. & Vacca, G. Mimetic finite difference methods for Hamiltonian wave equations in 2D, Computer and Matematics with applications, (2017), doi: 10.1016/j.camwa.2017.05.022.
Cacasús Acevedo, A. Aplicación del método de diferencias finitas generalizadas a problemas de elasto-dinámica. (Tesis inédita de doctorado), E.T.S. de Ingenieros Industriales UNED, Madrid, España, 2011.
De la Fuente, J.L. Técnicas de cálculo para sistemas de ecuaciones, programación lineal y programación entera: códigos en FORTRAN y C con aplicaciones de sistemas de energía eléctrica., Reverté, Madrid, España, 1997.
Fernández, L. A. Introducción a las ecuaciones en derivadas parciales, Universidad de Cantambria, Cantambria, España, 2016.
Gosse, L. Dirichlet-to-neumann mapping and finite-differences for anisotorpic difusion, Computers and Fluids, 156 (2017), 58–65.
Quintanilla Murillo, J. Métodos numéricos en diferencias finitas para la resolución de ecuaciones difusivas fraccionarias. (Tesis inédita de doctorado), Universidad de Extremadura, Badajoz, España, 2016.
Varga, R.S. Matrix iterative analysis., Prentice Hall, Englewood Cliffs, New Jersey, United State of America, 1962.
Wang, H., Liang, H., & Chai, Z. Finite-diference lattice Boltzmann model for nonlinear convection-diffusion equations, Applied Mathematics and Computation, 309 (2017), 334–349, doi: 10.1016/j.amc. 2017.04.015.
Xue, G. & Zhang, L. A new finite difference shemes for generalized Rosenau-Burgers equation, Applied Mathematics and Computation, 222(2013), 490–496, doi: 10.1016/j.amc.2013.07.052.
Zapata, M. U. & Itzá, R. High-order implicit finite difference shemes for the two dimensional Poisson equation, Applied Mathematics and Computation, 309(2017), 222–244, doi: 10.1016/j.1mc.2017.04.006.
Published
2019-06-29
How to Cite
Dı́az FerrerY., & Cruz Ramı́rezM. (2019). Algorithm for the numerical solution of the Poisson’s equation using the finite difference. Divulgaciones Matemáticas, 20(1), 67-77. Retrieved from https://produccioncientificaluz.org/index.php/divulgaciones/article/view/36622
Section
Research papers
