Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method

Eyaya Fekadie Anley

doi:doi:10.11648/j.pamj.20160504.16

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Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method

Eyaya Fekadie Anley

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4)

Received: 23 October 2015 Accepted: 16 November 2015 Published: 23 July 2016

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Abstract

Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 4)
DOI	10.11648/j.pamj.20160504.16
Page(s)	120-129
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Finite Volume Method, Discritization, PDEs, Control Volume (CV)

References

[1]	Herbin R. and O. Labergerie (2010), Finite volume schemes for elliptic and elliptic-hyperbolic Problems on triangular meshes, University of Chapman and Hall, Paris.
[2]	Klaus’s. 2010. Numerical Methods for Non-linear Elliptic Partial Differential Equation. Oxford University Press, New York.
[3]	Morton, K. W. and E. Suli, 2003. Finite volume methods and their analysis, IMA J. Numer.
[4]	Morton, K. W., Stynes M. and E. Suli (2005), Analysis of a cell-vertex finite volume method for a convection-diffusion problems, University of Albert, Canada.
[5]	Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society.
[6]	Holubová, Pavel Drábek; Gabriela (2007). Elements of partial differential equations.
[7]	Jost, J. (2002), Partial Differential Equations, New York: Springer-Verlag.
[8]	Lewy, Hans (1957), "An example of a smooth linear partial differential equation without solution.
[9]	Roubíček, T. (2013), Nonlinear Partial Differential Equations with Applications (2nd ed.).
[10]	Solin, P. (2005), Partial Differential Equations and the Finite Element Method, Hoboken, NJ: J. Wiley & Sons.
[11]	Solin, P.; Segeth, K. & Dolezel, I. (2003), Higher-Order Finite Element Methods, Boca Raton: Chapman & Hall/CRC Press.

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Eyaya Fekadie Anley. (2016). Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure and Applied Mathematics Journal, 5(4), 120-129. https://doi.org/10.11648/j.pamj.20160504.16

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ACS Style

Eyaya Fekadie Anley. Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure Appl. Math. J. 2016, 5(4), 120-129. doi: 10.11648/j.pamj.20160504.16

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AMA Style

Eyaya Fekadie Anley. Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure Appl Math J. 2016;5(4):120-129. doi: 10.11648/j.pamj.20160504.16

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@article{10.11648/j.pamj.20160504.16,
  author = {Eyaya Fekadie Anley},
  title = {Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {120-129},
  doi = {10.11648/j.pamj.20160504.16},
  url = {https://doi.org/10.11648/j.pamj.20160504.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.16},
  abstract = {Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.},
 year = {2016}
}

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TY  - JOUR
T1  - Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method
AU  - Eyaya Fekadie Anley
Y1  - 2016/07/23
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.16
DO  - 10.11648/j.pamj.20160504.16
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 129
PB  - Science Publishing Group
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AB  - Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.
VL  - 5
IS  - 4
ER  -

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Author Information

Eyaya Fekadie Anley

Department of Mathematics, College of Natural and Computational Science, School of Graduate Studies, Haramaya University, Haramaya, Ethiopia

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APA Style

Eyaya Fekadie Anley. (2016). Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure and Applied Mathematics Journal, 5(4), 120-129. https://doi.org/10.11648/j.pamj.20160504.16

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ACS Style

Eyaya Fekadie Anley. Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure Appl. Math. J. 2016, 5(4), 120-129. doi: 10.11648/j.pamj.20160504.16

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AMA Style

Eyaya Fekadie Anley. Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method. Pure Appl Math J. 2016;5(4):120-129. doi: 10.11648/j.pamj.20160504.16

Copy | Download

@article{10.11648/j.pamj.20160504.16,
  author = {Eyaya Fekadie Anley},
  title = {Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {120-129},
  doi = {10.11648/j.pamj.20160504.16},
  url = {https://doi.org/10.11648/j.pamj.20160504.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.16},
  abstract = {Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method
AU  - Eyaya Fekadie Anley
Y1  - 2016/07/23
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.16
DO  - 10.11648/j.pamj.20160504.16
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 129
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.16
AB  - Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.
VL  - 5
IS  - 4
ER  -

Copy | Download