A numerical method based on Crank-Nicolson scheme for Burgers’ equation Mohan. K. Kadalbajoo, A. Awasthi * Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208 016, India Abstract In this paper, we present a solution based on Crank-Nicolson finite difference method for one-dimensional Burgers’ equation. Burgers’ equation arises frequently in mathematical modeling of problems in fluid dynamics. Hopf-Cole trans- formation [E. Hopf, The partial differential equation u t + uu x = mu xx , Commun. Pure Appl. Math. 3 (1950) 201–230, J.D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951) 225–236] is used to linearize Burgers’ equation, the resulting heat equation is discretized by using Crank-Nicolson finite difference scheme. This method is shown to be unconditionally stable and second order accurate in space and time. Numerical results obtained by the present method have been compared with exact solution for different values of Reynolds’ number. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Burgers’ equation; Reynolds’ number; Half-Cole transformation; Crank-Nicolson finite difference method 1. Introduction We consider one-dimensional quasi-linear parabolic partial differential equation, ou ot þ u ou ox ¼ 1 Re o 2 u ox 2 ; ðx; tÞ2 X; ð1:1aÞ where X ¼ð0; 1Þð0; T ; with initial condition uðx; 0Þ¼ f ðxÞ; 0 < x < 1; ð1:1bÞ and boundary conditions 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.030 * Corresponding author. E-mail addresses: kadal@iitk.ac.in (Mohan. K. Kadalbajoo), ashishan@iitk.ac.in (A. Awasthi). Applied Mathematics and Computation 182 (2006) 1430–1442 www.elsevier.com/locate/amc