Numerical Solutions of Advection-Diffusion Equation via Quartic B-Spline Least Squares Method

Year 2023, Volume: 4 Issue: 3, 9 - 14, 27.12.2023

Buket Ay Bülent Saka Ozlem Ersoy Hepson İdris Dağ

https://doi.org/10.53608/estudambilisim.1310048

Abstract

Numerical solutions are obtained for the time dependent one dimensional Advection-Diffusion equation (ADE) using the least squares method (LSM). After the time-integration of the ADE by use of the Crank-Nicolson technique, the combination of the quartic B-splines as an trial function is substituted in the time discretized ADE and for the LSM procedure, a weight function is obtained by taking derivative of the integrand of integral of squared ADE with respect to time parameters of the trial functions. Thus solution of obtained recursive system of the unknown parameters provides solutions of the ADE at discrete times. Achievement is displayed by studying three test problems.

Keywords

Advection-Diffusion Denklemi, B-splines, Least Squares Method

Adveksiyon Difüzyon Denkleminin Kuartik B-Spline En Küçük Kareler Metodu ile Sayısal Çözümü

Abstract

En küçük Kareler metodu (LSM) kullanılarak zamana bağlı bir boyutlu Advection-Diffusion denklemi (ADE) için yaklaşık çözümler elde edilmiştir. Crank-Nicolson yöntemi ile ADE’in zaman ayrıştırması yapıldıktan sonra, Kuadratik B-spline fonksiyonların kombinasyonundan oluşan deneme fonksiyonu zaman ayrışmış ADE’nin bilinmeyen fonksiyonuna yerleştirilmiştir. Böylece elde edilen cebirsel ifadenin karesi alınmış integralinin integrantının deneme fonksiyonun da tanımlı zaman parametresine göre türevi alınarak ağırlık fonksiyonu belirlenmiştir. Bulunan iterative sistemin çözümlerinden ADE’nin ayrık zamanlardaki çözümlerini elde edilir. Metodun başarısı üç test problem çalışılarak gösterilmiştir.

Keywords

Adveksiyon-Difuzyon Denklemi, B-splines, En Küçük Kareler Yöntemi

References

• [1] Kirli, E. 2023. Quintic Trigonometric B-spline Algorithm for Numerical Solution of the Modified Regularized Long Wave Equation. ESTUDAM Bilişim Derg, 4(2), 10-15.
• [2] Dag,I., Canıvar, A., Şahin, A. 2011. Taylor-Galerkin method for advection-diffusion equation. Kybernetes, 40, 762-777.
• [3] Mittal,R. C., Jain, R. K. , 2012. Redefned cubic B-spline collocation method for solving convection diffusion equations. Appl. Math. Model. 36, 5555–5573.
• [4] Mittal, R.C., Rohila, R. 2020. The numerical study of advection–difusion equations by the fourth-order cubic B-spline collocation method. Mathematical Sciences 14, 409–423.
• [5] Korkmaz, I., Dag, A. 2012. Cubic B-spline differential quadrature methods for the advection-diffusion equation. International Journal of Numerical Methods for Heat and Fluid Flow. 22(8), 1021-1036.
• [6] Irk ,D., Dag, I., Tombul, M. 2015. Extended Cubic B-Spline Solution of the Advection-Diffusion Equation. KSCE J. Civ. Eng., 19, 929-934.
• [7] Korkmaz , A., Dag, I. 2016. Quartic and quintic B-spline methods for advection diffusion equation. Appl. Math. Comput., 274, 208-219.
• [8] Görgülü, M. Z., Dağ, İ., Doğan, S., Irk, D. 2018. A numerical solution of the Advection-Diffusion Equation by using extended cubic B-spline functions. Anadolu Univ. J. of Sci. and Technology A -Appl. Sci. and Eng. 2 (19), 347-355.
• [9] Görgülü, M. Z., Irk, D. 2019. The Galerkin Finite Element Method for advection diffusion equation. Sigma Journal of Engineering and Natural Science 37(1), 119-128.
• [10] Yousaf, A., Abdeljawad, T., Yaseen, M., Abbas, M. 2020. Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation. Mathematical Problems in Engineering, Article ID 8908964, 17 pages.
• [11] Nguyen, H., Reynen, J. 1984. A spacetime least-squares finite elements scheme for advection-diffusion equation. Comput. Methods Appl. Mech. Eng. 42, 331-442.
• [12] Kadalbajoo M. K., Arora, P. 2010. Space–time Galerkin least-squares method for the one–dimensional Advection–Diffusion Equation. International Journal of Computer Mathematics 87(1), 103-118.Shin-Jye, L. 1997. p-Version space-time least-squares finite-element method for unsteady Convection-Diffusion and Burgers’ Equations. IEEE, 1, 277-282.
• [13] Dhawan, S., Bhowmik, S., Kumar, K. S. 2015. Galerkin-least square B-spline approach toward advection diffusion equation. Applied Mathematics and Computation, 261, 128-140.
• [14] Dag, I., Irk, D., Tombul, M. 2006. Least-squares finite element method for the Advection Diffusion Equation. Appl. Math. Comput., 173, 554-565, 2006.
• [15] Shin-Jye, L. 1997. p-Version space-time least-squares finite-element method for unsteady Convection-Diffusion and Burgers’ Equations. IEEE, 1, 277-282.
• [16] Dhawan,S., Kapoor, S., Kumar, S. 2012. Numerical method for Advection Diffusion Equation using FEM and B-splines. Journal of Computational Science 3, 429-437.
• [17] Dag, I., Ay B., Saka B. 2022. Cubic B-spline least squares method for the numerical solution of advection-diffusion. J. Appl. Comp. Sci. equation, 1(1), 53-58.
• [18] Kirli, E. 2023. Nonic B-spline Approach for Advection-Diffusion Equation. Eskişehir Technical Univ. J. of Sci. and Tech. A-Appl. Sci. and Eng, 24(2), 155-163.