| Peer-Reviewed

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

Received: 23 October 2015     Accepted: 16 November 2015     Published: 23 July 2016
Views:       Downloads:
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), 2016. 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.
Cite This Article
  • 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

    Copy | Download

    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

    Copy | Download

    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

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

  • Sections