Numerical simulation of Burgers’ equation using cubic B-splines

Abstract

In this paper, a numerical θ scheme is proposed for solving nonlinear Burgers’ equation. By employing Hopf-Cole transformation, the nonlinear Burgers’ equation is linearized to the linear Heat equation. The resulting Heat equation is further solved by cubic B-splines. The time discretization of linear Heat equation is carried out using Crank-Nicolson scheme θ=12$\left( {\theta = {\textstyle{1 \over 2}}} \right)$ as well as backward Euler scheme (θ = 1). Accuracy in temporal direction is improved by using Richardson extrapolation. This method hence possesses fourth order accuracy both in space and time. The system of matrix which arises by using cubic splines is always diagonal. Therefore, working with splines has the advantage of reduced computational cost and easy implementation. Stability of the schemes have been discussed in detail and shown to be unconditionally stable. Three examples have been examined and the L2 and L∞ error norms have been calculated to establish the performance of the method. The numerical results obtained on applying this method have shown to give more accurate results than existing works of Kutluay et al. [1], Ozis et al. [2], Dag et al. [3], Salkuyeh et al. [4] and Korkmaz et al. [5].

Keywords

• burgers’ equation
• hopf-cole transformation
• cubic b-spline
• crank-nicolson scheme
• backward euler scheme
• richardson extrapolation