PRAMANA
c Indian Academy of Sciences
— journal of physics
Vol. 81, No. 4 October 2013 pp. 547–556
Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods BILGE INAN∗ and AHMET REFIK BAHADIR Department of Mathematics, Faculty of Arts and Science, Inonu University, 44280 Malatya, Turkey ∗ Corresponding author. E-mail:
[email protected] MS received 14 February 2013; revised 24 June 2013; accepted 4 July 2013 DOI: 10.1007/s12043-013-0599-z; ePublication: 21 September 2013 Abstract. This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable. Keywords. Burgers’ equation; exponential finite difference method; implicit exponential finite difference method; fully implicit exponential finite difference method. PACS Nos 02.60.−x; 02.30.Jr; 02.70.Bf
1. Introduction In this paper, we consider the one-dimensional non-linear Burgers’ equation ∂u ∂ 2u ∂u +u − v 2 = 0, ∂t ∂x ∂x with the initial condition u (x, 0) = g(x),
a < x < b,
(1)
a
