Higher Order Accurate Numerical Solution of Advection Diffusion Equation

Yıl 2018, Cilt: 6 Sayı: 2, 253 - 258, 15.10.2018

Dursun Irk Melis Zorşahin Görgülü

Öz

In this study, the advection diffusion equation (ADE) will be solved numerically using the quintic B-spline Galerkin finite-element method, based on second and fourth order single step methods for time integration. Two test problems are studied and accuracy of the numerical results are measured by the computing the order of convergence and error norm $L_{\infty }$ for the proposed methods. The numerical results of this study demonstrate that the proposed two algorithms especially the fourth order single step method are a remarkably successful numerical technique for solving the advection diffusion equation.

Anahtar Kelimeler

Galerkin method, Quintic B-spline, Taylor series

Kaynakça

• [1] M. Dehghan, Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Comput., 147 (2004), 307-319.
• [2] M. Sari , G. G¨uraslan and A. Zeytinoglu, High-Order finite difference schemes for solving the advection-diffusion equation, Math. Comput. Appl., 15 (2010), 449-460.
• [3] I. Da˘g, D. Irk and M. Tombul, Least-squares finite element method for the advection diffusion equation, Appl. Math. Comput., 173 (2006), 554-565.
• [4] I. Da˘g, A. Canıvar and A. S¸ ahin, Taylor-Galerkin method for advection-diffusion equation, Kybernetes, 40 (2011), 762-777.
• [5] D. Irk, ˙I. Da˘g and M. Tombul, Extended Cubic B-Spline Solution of the Advection-Diffusion Equation, KSCE J. Civ. Eng., 19(2015), 929-934.
• [6] A. Korkmaz and ˙I. Da˘g, Quartic and quintic B-spline methods for advection diffusion equation, Appl. Math. Comput., 274 (2016), 208-219.
• [7] R.C. Mittal and G. Arora, Quintic B-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation, Commun. Nonlinear. Sci., 15 (2010), 2798-2808.
• [8] S.S. Siddiqi and S. Arshed, Quintic B-spline for the numerical solution of the good Boussinesq equation, Journal of the Egyptian Mathematical Society, 22 (2014), 209-213.
• [9] B. Saka, A quintic B-spline finite-element method for solving the nonlinear Schr¨odinger equation, Phys. Wawe Phenom. 20 (2012), 107-117.
• [10] A. Bas¸han, S.B.G. Karakoc¸ and T. Geyikli, Approximation of the KdVB equation by the quintic B-spline differential quadrature method, Kuwait J. Sci., 42 (2015), 67-92.
• [11] P.M. Prenter, Splines and variational methods, J. Wiley, 1975.