A Robust Method for the Numerical Solution of the Advection-Diffusion Equation Using the Galerkin Method Based on Hermite B-Spline Functions

Abstract

The goal of this research is to find an approximate solution to the one-dimensional linear advection-diffusion equation using the cubic Hermite B-Spline Galerkin finite element method, a new numerical method developed by selecting cubic Hermite B-spline functions as the basis functions. After discretizing the advection-diffusion equation in the temporal direction using the Crank-Nicolson formula, the spatial discretization of the semi-discrete scheme was performed using cubic Hermite B-spline functions. The effectiveness and accuracy of the proposed method were tested using the four most well-known model problems in the literature. The obtained numerical results were compared with those of other studies in the literature and presented in tabular form. Additionally, to observe the agreement of the approximate solutions with the analytical solution of the model problems, the graphs of the analytical and the approximate solutions were plotted together, and the graphs of the absolute error were plotted.

Keywords

• Galerkin method
• Advection-diffusion equation
• Hermite B-spline function
• Finite element method
• Legendre polynomials

Ethical Statement

Ethics committee approval was not required for this study because of there was no study on animals or humans.

References

1. Başhan, A., Uçar, Y., Yağmurlu, N.M & Esen, A. (2018). A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation. The European Physical Journal Plus, 133(1), 12.
2. Başhan, A., Uçar, Y., Yağmurlu, N.M & Esen, A. (2020). Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation. Numerical Methods for Partial Differential Equations, 37, 690–706.
3. Cecchi, M. M., & Pirozzi, M. A. (2005). High order finite difference numerical methods for time-dependent convection-dominated problems. Applied Numerical Mathematics, 55(3), 334–356.
4. Chapman, S. J., Shipley, R. J., & Jawad, R. (2008). Multiscale modeling of fluid transport in tumors. Bulletin of Mathematical Biology, 70(8), 2334–2357.
5. Chatwin P. C., & Allen, C. M. (1985). Mathematical models of dispersion in rivers and estuaries. Annual Review of Fluid Mechanics, 17(1), 119–149.
6. Chawla, M. M., Al-Zanaidi, M. A., & Al-Aslab, M. G. (2000). Extended one-step time-integration schemes for convection-diffusion equations. Computers & Mathematics with Applications, 39(3–4), 71–84.
7. Dağ, İ., Canivar, A., & Şahin, A. (2011). Taylor‐Galerkin method for advection‐diffusion equation. Kybernetes, 40(5–6), 762–777.
8. Dehghan, M. (2004). Weighted finite difference techniques for the one-dimensional advection–diffusion equation. Applied Mathematics and Computation, 147(2), 307–319.

9. Ganaie I. A., & Kukreja, V. K. (2014). Numerical solution of Burgers’ equation by cubic Hermite collocation method. Applied Mathematics and Computation, 237, 571–581.
10. Goh, J., Majid, M. A., & Ismail, A. I. M. (2012). Cubic B-spline collocation method for one-dimensional heat and advection-diffusion equations. Journal of Applied Mathematics, 2012.
11. Guvanasen V., & Volker, R. E. (1983). Numerical solutions for solute transport in unconfined aquifers. International Journal for Numerical Methods in Fluids, 3(2), 103–123.
12. Iqbal, A., Akram, T., Khan, W. A., & Ali, A. (2023). A Powerful High-Order B-Spline Galerkin Technique for Numerical Solutions of Advection-Diffusion Equation. International Conference on Soft Computing: Theories and Applications, 59–68.
13. Isenberg J., & Gutfinger, C. (1973). Heat transfer to a draining film. International Journal of Heat and Mass Transfer, 16(2), 505–512.
14. Kadalbajoo M. K., & Arora, P. (2010). Taylor-Galerkin B-spline finite element method for the one-dimensional advection-diffusion equation. Numerical Methods for Partial Differential Equations, 26(5), 1206–1223.
15. Karahan, H. (2007). Unconditional stable explicit finite difference technique for the advection-diffusion equation using spreadsheets. Advances in Engineering Software, 38(2), 80–86.
16. Korkmaz, A., & Dağ, I. (2016). Quartic and quintic B-spline methods for advection–diffusion equation. Applied Mathematics and Computation, 274, 208–219. https://doi.org/10.1016/J.AMC.2015.11.004
17. Mittal, A. K., Ganaie, I. A., Kukreja, V. K., Parumasur, N., & Singh, P. (2013). Solution of diffusion-dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis. Computers and Chemical Engineering, 58, 203–210.
18. Mittal, R. C., & Jain, R. K. (2012). Redefined cubic B-splines collocation method for solving convection–diffusion equations. Applied Mathematical Modelling, 36(11), 5555–5573.
19. Mittal, R. C., & Rohila, R. (2020). The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method. Mathematical Sciences, 14(4), 409–423.
20. Mohammadi, R. (2013). Exponential B-spline solution of convection-diffusion equations. Applied Mathematics, 4(6), 933–944.
21. Mohebbi A., & Dehghan, M. (2010). High-order compact solution of the one-dimensional heat and advection-diffusion equations. Applied Mathematical Modelling, 34(10), 3071–3084.
22. Naghipoor, J., Jafary, N., & Rabczuk, T. (2018). Mathematical and computational modeling of drug release from an ocular iontophoretic drug delivery device. International Journal of Heat and Mass Transfer, 123, 1035–1049.
23. Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A., & Rashid, A. (2016). The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Applied Mathematical Modelling.
24. Novikau, I., & Joseph, I. (2025). Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems. Computer Physics Communications, 309, 109498.
25. Palitta, D., & Simoncini, V. (2016). Matrix-equation-based strategies for convection–diffusion equations. BIT Numerical Mathematics, 56(2), 751–776.
26. Parlange, J. Y. (1980). Water transport in soils. Annual Review of Fluid Mechanics, 12, 77–102.
27. Prenter, P. M. (1989). Splines and variational methods. Wiley.
28. Rizwan-Uddin. (1997). A second-order space and time nodal method for the one-dimensional convection-diffusion equation. Computers & Fluids, 26(3), 233–247.
29. Rohila, R. (2024). Numerical Solution of Advection-Diffusion Equation by a Modified Cubic B-Spline Collocation Method. In Tuijin Jishu/Journal of Propulsion Technology, 45(3), 4297 – 4313.
30. Salkuyeh, D. K. (2006). On the finite difference approximation to the convection-diffusion equation. Applied Mathematics and Computation, 179(1), 79–86.
31. Ünal, O. (2022). A Stable and High Accurate Numerical Simulator for Convection-Diffusion Equation. Journal of Advanced Research in Natural and Applied Sciences, 8(1), 45–53.
32. Zlatev, Z., Berkowicz, R., & Prahm, L. P. (1984). Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants. Journal of Computational Physics, 55(2), 278–301.