Research ArticleOpen AccessGoogle Scholar indexed
Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI
Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
- 1 Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
- 2 Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
- 3 Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia
American Journal of Computational Mathematics·Volume 07 (2017)·Pages 208–227·Published 1 August 2017·DOI10.4236/ajcm.2017.73019
Copy link · social · email
Abstract
To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L 2 , L<sub> ∞</sub> norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.
KeywordsCrank-NicholsonTaylor’s SeriesNewton’s Iterative MethodAlternating Direction Implicit (ADI)
- Pandey, K.P., Lajja, V.L. and Amit, K.V. (2009) On a Finite Difference Scheme for Burgers Equation. Applied Mathematics and Computation, 215, 2208-2214. https://doi.org/10.1016/j.amc.2009.08.018
- Fletcher, C.A. (1983) Generating Exact Solutions of the Two-Dimensional Burgers Equations. International Journal for Numerical Methods in Fluids, 3, 213-216. https://doi.org/10.1002/fld.1650030302
- Collatz, L.C. (1960) The Numerical Treatment of Differential Equations. Springer Verlag, Berlin, 28. https://doi.org/10.1007/978-3-662-05500-7
- Zhu, H.Q., Shu, H.Z. and Ding, M.Y. (2010) Numerical Solution of Two-Dimensional Burgers’ Equations by Discrete Adomian Decomposition Method. Computers and Mathematics with Applications, 60, 840-848. https://doi.org/10.1016/j.camwa.2010.05.031
- Mazumder, S.M. (2015) Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods. Academic Press, New York.
- Bateman, H.B. (1915) Some Recent Researches on the Motion of Fluids. Monthly Weather Review, 43, 163-170. https://doi.org/10.1175/1520-0493(1915)43 2.0.CO;2
- Crandall, S.H. (1956) Engineering Analysis. McGraw-Hill, New York, 151.
- Noye, J.N. (1981) Nonlinear Partial Differential Equations in Engineering. University of Melbourne, Melbourne.
- Schiesser, W.E. and Griffiths, G.W. (2009) A Compendium of Partial Differential Equation Models. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511576270
- Srivastava, V.K. and Tamsir, M.T. (2012) Crank Nicolson Semi Implicit Approach for Numerical Solution of Two Dimensional Coupled Non-Linear Burgers Equations. International Journal of Applied Mechanics and Engineering, 17, 571-581.
- Ames, W.F. (1965) Non-Linear Partial Differential Equations in Engineering. Academic Press, New York.
- Ames, W.F. (1969) Finite Difference Methods for Partial Differential Equations. Academic Press, New York.
- James, E.F. (2002) An Introduction to Numerical Analysis. John Wiley and Sons, New York.
- Roache, P.J. (1972) Computational Fluid Dynamics. Hermosa, Albuquerque.
- Mohammad, T.M. and Vineet, K.S. (2011) A Semi-Implicit Finite-Difference Approach for Two-Dimensional Coupled Burgers Equations. International Journal of Scientific and Engineering Research, 2, 9-18.