Abstract
In physics and mathematics, heat equation is a special case of diffusion equation and is a partial differential equation (PDE). Partial differential equations are useful tools for mathematical modeling. A few problems can be solved analytically, whereas difficult boundary value problem can be solved by numerical methods easily. A very popular numerical method known as finite difference methods (explicit and implicit schemes) is applied expansively for solving heat equations successfully. Explicit schemes are Forward Time and Centre Space (FTCS), Dufort and Frankel methods, whereas implicit schemes are Laasonen and Crank-Nicolson methods. In this study, explicit and implicit finite difference schemes are applied for simple one-dimensional transient heat conduction equation with Dirichlet’s initial-boundary conditions. MATLAB code is used to solve the problem for each scheme in fine mesh grids. Comparing results with analytical results, Crank-Nicolson method gives the best approximate solution. FTCS scheme is conditionally stable, whereas other schemes are unconditionally stable. Convergence, stability and truncation error analysis are investigated. Transient temperature distribution plot and surface temperature plots for different time are presented. Also, unstable plot for FTCS method is represented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ames, W.F.: Numerical Methods for Partial Differential Equations, 3rd edn. Academic Press Inc., Boston (1992)
Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations: An Introduction. Cambridge University Press, Cambridge, England (1994)
Cooper, J.: Introduction to Partial Differential Equations with Matlab. Birkhauser, Boston (1998)
Clive, A.J.F.: Computational Techniques for Fluid Dynamics. Springer-Verlag, Berlin (1988)
Golub, G., Ortega, J.M.: Scientific Computing: An Introduction with Parallel Computing. Academic Press, Inc., Boston (1993)
Burden, R.L., Faires, J.D.: Numerical Analysis, 6th edn. Brooks/Cole Publishing Co., New York (1997)
Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods, vol. 22. Springer Science & Business Media (2013)
Strikwerda, J.C.: Finite difference schemes and partial differential equations. SIAM J. Appl. Sci. Environ. Sustain. 3(7), 188–200 (2004). e-ISSN 2360-8013 200 | P a g e
Arnold, D.N.: Lecture Notes on Numerical Analysis of Partial Differential Equations. Available at http://www.math.umn.edu/~arnold/8445/notes.pdf (2015)
Trefethen, L.N.: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Unpublished Text. Available at https://people.maths.ox.ac.uk/trefethen/pdetext.html (1996)
Olsen-Kettle, L.: Numerical Solution of Partial Differential Equations. Retrieved from http://espace.library.uq.edu.au/view/UQ:239427 (2015)
Ciegis, R.: Numerical solution of hyperbolic heat conduction equation. Math. Model. Anal. 14(1), 11–24 (2009)
Recktenwald, G.W.: Finite-Difference Approximations to the Heat Equation. http://www.nada.kth.se/~jjalap/numme/FDheat.pdf (2011)
Karatay, I., Kale, N., Bayramoglu, S.: A new difference scheme for time fractional heat equations based on the Crank-Nicholson method. Fract. Calc. Appl. Anal. 16(4), 892–910 (2013)
Aswin, V.S., Awasthi, A., Anu, C.: A comparative study of numerical schemes for convection-diffusion equation. Procedia Eng. 127, 621–627 (2015)
Azad, T.M.A.K., Andallah, L.S.: Stability analysis of finite difference schemes for an advection diffusion equation. Bangladesh J. Sci. Res. 29(2), 143–151 (2016)
Mebrate, B.: Numerical solution of one dimensional heat equation with Dirichlet boundary condition. Am. J. Appl. Math. 3(6), 305–311 (2015)
Adak, M., Mandal, N.R.: Numerical and experimental study of mitigation of welding distortion. Appl. Math. Model. 34, 146–158 (2010)
Adak, M., Soares, C.G.: Effects of different restraints on the weld-induced residual deformations and stresses in a steel plate. Int. J. Adv. Manuf. Technol. 71, 699–710 (2014)
Adak, M., Soares, C.G.: Residual deflections and stresses in a thick T joint plate structure. J. Appl. Mech. Eng. 5, 6 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Adak, M. (2020). Comparison of Explicit and Implicit Finite Difference Schemes on Diffusion Equation. In: Bhattacharyya, S., Kumar, J., Ghoshal, K. (eds) Mathematical Modeling and Computational Tools. ICACM 2018. Springer Proceedings in Mathematics & Statistics, vol 320. Springer, Singapore. https://doi.org/10.1007/978-981-15-3615-1_15
Download citation
DOI: https://doi.org/10.1007/978-981-15-3615-1_15
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-3614-4
Online ISBN: 978-981-15-3615-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)