Application of Some Finite Difference Schemes for Solving One Dimensional Diffusion Equation
Keywords:
Crank – Nicolson, Diffusion equation, Forward time centered space, Backward time centered space, Stability.Abstract
In this paper the numerical solutions of one dimensional diffusion equation using some finite difference methods have been considered. For that purpose three examples of the diffusion equation together with different boundary conditions are examined. The finite difference methods applied on each example are (i) forward time centered space (ii) backward time centered space and (iii) Crank – Nicolson. In each case, we have studied stability of finite difference method and also obtained numerical result. The performance of each scheme is evaluated in accordance with both the accuracy of the solution and programming efforts. The implementation and behavior of the schemes have been compared and the results are illustrated pictorially. It is found in case of the test examples studied here that the Crank – Nicolson scheme gives better approximations than the two other schemes.
References
[2] Evans G., Blackdge J. and Yardley P. (2000). Numerical Methods for Partial Differential Equations, Springer – Verlag, Berlin, Heidelberg, New York. ISBN 3-540-76125-X.
[3] Gerald W. (2011). Finite – Difference Approximations to the Heat Equation, Portland State University, Portland, Oregon.
[4] H. Saberi Najafi (2008). Solving One – Dimensional Advection – Dispersion with Reaction Using Some Finite – Difference Methods, Applied Mathematical Sciences, Vol. 2, Pp. 2611 – 18.
[5] Karantonis A. (2001). Numerical Solution of Reaction – Diffusion Equations by the Finite Difference Method.
[6] Kværnø A. (2009). Numerical Mathematics, Lecture Notes, NTNU, University of Oslo.
[7] Owren B. (2011). Numerical Solution of Partial Differential Equations with Finite Difference Methods, Lecture Notes, NTNU, University of Oslo.
[8] Langtangen H. P. (2013). Finite difference methods for diffusion processes, University of Oslo.
[9] Shahraiyni Taheri H. and Ataie B. (2009). Comparison of Finite Difference Schemes for Water Flow in Unsaturated Soils, International Journal of Aerospace and Mechanical Engineering, Vol. 3(1), Pp.1 – 5.
[10] Sweilam N. H., Khader M. M. and Mahdy A. M. S. (2012). Crank – Nicolson Finite Difference Method for Solving Time – Fractional Diffusion Equation, Journal of Fractional Calculus and Applications, Vol. 2, No. 2, Pp. 1 – 9, http://www.fcaj.webs.com
[11] Thongmoon M. and McKibbin R. (2006). A Comparison of Some Numerical Methods for the Advection – Diffusion Equation, Res. Lett. Inf. Math. Sci., Vol. 10, Pp 49 – 62. http://iims.massey.ac.nz/research/letters/
[12] Tsegaye Simon (2013). Numerical Simulation of Diffusion – Reaction Equations: Application from River Pollution Model. Hawassa University, Hawassa, Ethiopia (Unpublished M. Sc. Thesis).
[13] Won Y., Wenwu C., Tae – Sang C. and John M. (2005). Applied Numerical Methods Using MATLAB, ISBN 0-471-69833-4 (cloth), John Wiley & Sons, Inc., Hoboken, New Jersey.
[14] Zana S. (2014). Numerical Solution of Diffusion Equation in One Dimension, Eastern Mediterranean University, Gazima?usa, North Cyprus.
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