Sextic B-spline Galerkin method for advection diffusion equation

Yıl 2018, Cilt: 20 Sayı: 3, 105 - 116, 29.10.2018

Evren Topcu , Dursun Irk

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

Öz

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.

Anahtar Kelimeler

Advection diffusion equation, sextic B-spline, Galerkin method

Adveksiyon difüzyon denklemi için sektik B-spline Galerkin metodu

Öz

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.

Anahtar Kelimeler

Adveksiyon difüzyon denklemi, Sektik B-spline, Galerkin yöntemi

Kaynakça

• 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).