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Sextic B-spline Galerkin method for advection diffusion equation

Year 2018, Volume: 20 Issue: 3, 105 - 116, 29.10.2018

Evren Topcu , Dursun Irk

https://doi.org/10.25092/baunfbed.481169

Abstract

In this study, sextic B-spline Galerkin finite element method is proposed for numerical solution of the advection diffusion equation.  In the method, second, three and fourth order single step methods are used for the time integration.  Second order single step method is also known as Crank Nicolson method.  Two numerical examples are studied to illustrate the accuracy and the efficiency of the proposed methods.

Keywords

Advection diffusion equation sextic B-spline Galerkin method

References

  • Karur, S.R. and Ramachandran, P.A., Augmented Thin Plate Spline Approximation in DRM, Boundary Elements Communications, 6, 55-58.(1995).
  • Dehghan M., Weighted finite difference techniques for the one-dimensional advection-diffusionequation, Applied Mathematics and Computation, 147, 307-319 (2004).
  • Sari M., Güraslan G. and Zeytinoglu A , High-Order finite difference schemes for solving the advection-diffusion equation, Mathematical and Computational Applications, 15 (3), 449-460 (2010).
  • Dağ I., Irk D. and Tombul M., Least-squares finite element method for the advection diffusion equation, Applied Mathematics and Computation, 173, 554-565 (2006).
  • Kapoor, S. and Dhawan, S., B-spline finite element technique for advection diffusion equation. International Journal of Applied Mathematics and Mechanics, 6, 75-94 (2010).
  • Dağ I. Canıvar A. and ¸Sahin A., Taylor-Galerkin method for advection-diffusion equation, Kybernetes , 40, 762-777 (2011).
  • Dhawan, S., Kapoor, S. and Kumar, S., Numerical method for advection diffusion equation using FEM and B-splines, Journal of Computational Science 3,429-443 ( 2012).
  • Irk D., Dağ I. and Tombul M., Extended Cubic B-Spline Solution of the Advection-Diffusion Equation, KSCE Journal of Civil Engineering, 19(4), 929-934 (2015).
  • Clough R.W., The finite element method in plane stress analysis, Proc. 2nd Conf. On Electronic Computation, Pittsburgh (1960).
  • Ritz W. Über eine neue methode zur lösung gewisser variations probleme der mathematischen physik, J. Reine Angew. Math., 135, 1-61, (1908).
  • Rayleigh J.W.S. On the theory of resonance, Trans. Roy. Soc. (London), A161 77-118, (1870).
  • Rayleigh J.W.S. The Theory of Sound, Dover Publications, 2nd Editon, (1945).
  • Galerkin B.G., Stabe und Platten; Reihen in Gewissen Gleichgewichtspoblemen Elestischer Stabe und Platten, Vestnik der Ingenieure, 19 897-908, (1915).
  • Zienkiewicz O.C and Morgan K., Finite Element and Approximation, John Wiley & Sons, (1983).
  • Mohammadi R, Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models, Applied Mathematics and Computation, 241, 151-166, (2014)
  • Irk D., Sextic B-spline collocation method for the modified burgers' equation, Kybernetes, 38(9) 1599-1620, (2009).

Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu

Year 2018, Volume: 20 Issue: 3, 105 - 116, 29.10.2018

Evren Topcu , Dursun Irk

https://doi.org/10.25092/baunfbed.481169

Abstract

Bu çalışmada sektik B-spline Galerkin metodu adveksiyon difüzyon denkleminin yaklaşık çözümü için önerilmiştir.  Önerilen metotta zaman parçalanması için doğruluğu iki, üç ve dört olan tek adımlı yöntemler kullanılmıştır.  Doğruluğu iki olan yöntem Crank-Nicolson yöntemi olarak ta bilinmektedir.  İki sayısal örnek kullanılarak önerilen yöntemlerin etkinliği ve doğruluğu kontrol edilmiştir.

Keywords

Adveksiyon difüzyon denklemi Sektik B-spline Galerkin yöntemi

References

  • Karur, S.R. and Ramachandran, P.A., Augmented Thin Plate Spline Approximation in DRM, Boundary Elements Communications, 6, 55-58.(1995).
  • Dehghan M., Weighted finite difference techniques for the one-dimensional advection-diffusionequation, Applied Mathematics and Computation, 147, 307-319 (2004).
  • Sari M., Güraslan G. and Zeytinoglu A , High-Order finite difference schemes for solving the advection-diffusion equation, Mathematical and Computational Applications, 15 (3), 449-460 (2010).
  • Dağ I., Irk D. and Tombul M., Least-squares finite element method for the advection diffusion equation, Applied Mathematics and Computation, 173, 554-565 (2006).
  • Kapoor, S. and Dhawan, S., B-spline finite element technique for advection diffusion equation. International Journal of Applied Mathematics and Mechanics, 6, 75-94 (2010).
  • Dağ I. Canıvar A. and ¸Sahin A., Taylor-Galerkin method for advection-diffusion equation, Kybernetes , 40, 762-777 (2011).
  • Dhawan, S., Kapoor, S. and Kumar, S., Numerical method for advection diffusion equation using FEM and B-splines, Journal of Computational Science 3,429-443 ( 2012).
  • Irk D., Dağ I. and Tombul M., Extended Cubic B-Spline Solution of the Advection-Diffusion Equation, KSCE Journal of Civil Engineering, 19(4), 929-934 (2015).
  • Clough R.W., The finite element method in plane stress analysis, Proc. 2nd Conf. On Electronic Computation, Pittsburgh (1960).
  • Ritz W. Über eine neue methode zur lösung gewisser variations probleme der mathematischen physik, J. Reine Angew. Math., 135, 1-61, (1908).
  • Rayleigh J.W.S. On the theory of resonance, Trans. Roy. Soc. (London), A161 77-118, (1870).
  • Rayleigh J.W.S. The Theory of Sound, Dover Publications, 2nd Editon, (1945).
  • Galerkin B.G., Stabe und Platten; Reihen in Gewissen Gleichgewichtspoblemen Elestischer Stabe und Platten, Vestnik der Ingenieure, 19 897-908, (1915).
  • Zienkiewicz O.C and Morgan K., Finite Element and Approximation, John Wiley & Sons, (1983).
  • Mohammadi R, Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models, Applied Mathematics and Computation, 241, 151-166, (2014)
  • Irk D., Sextic B-spline collocation method for the modified burgers' equation, Kybernetes, 38(9) 1599-1620, (2009).
There are 16 citations in total.

Details

Primary Language Turkish
Journal Section Research Articles
Authors

Evren Topcu 0000-0002-8087-7130

Dursun Irk 0000-0002-3340-1578

Publication Date October 29, 2018
Submission Date August 19, 2018
Published in Issue Year 2018 Volume: 20 Issue: 3

Cite

APA Topcu, E., & Irk, D. (2018). Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(3), 105-116. https://doi.org/10.25092/baunfbed.481169
AMA Topcu E, Irk D. Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu. BAUN Fen. Bil. Enst. Dergisi. October 2018;20(3):105-116. doi:10.25092/baunfbed.481169
Chicago Topcu, Evren, and Dursun Irk. “Adveksiyon difüzyon Denklemi için Sektik B-Spline Galerkin Metodu”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20, no. 3 (October 2018): 105-16. https://doi.org/10.25092/baunfbed.481169.
EndNote Topcu E, Irk D (October 1, 2018) Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20 3 105–116.
IEEE E. Topcu and D. Irk, “Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu”, BAUN Fen. Bil. Enst. Dergisi, vol. 20, no. 3, pp. 105–116, 2018, doi: 10.25092/baunfbed.481169.
ISNAD Topcu, Evren - Irk, Dursun. “Adveksiyon difüzyon Denklemi için Sektik B-Spline Galerkin Metodu”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20/3 (October 2018), 105-116. https://doi.org/10.25092/baunfbed.481169.
JAMA Topcu E, Irk D. Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu. BAUN Fen. Bil. Enst. Dergisi. 2018;20:105–116.
MLA Topcu, Evren and Dursun Irk. “Adveksiyon difüzyon Denklemi için Sektik B-Spline Galerkin Metodu”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 20, no. 3, 2018, pp. 105-16, doi:10.25092/baunfbed.481169.
Vancouver Topcu E, Irk D. Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu. BAUN Fen. Bil. Enst. Dergisi. 2018;20(3):105-16.