A Higher Order Numerical Implicit Method for Non-Linear Burgers’ Equation

Abstract

This paper proposes a higher order implicit numerical scheme to approximate the solution of the nonlinear partial differential equation (PDE). This equation is a simplified form of Navier–Stoke’s equation also known as Burgers’ equation. It is an important nonlinear PDE which arises frequently in mathematical modeling of turbulence in fluid dynamics. In order to handle nonlinearity a nonlinear transformation is used which converts the nonlinear PDE into a linear PDE. The linear PDE is semi-discretized in space by method of lines to yield a system of ordinary differential equations in time. The resulting system of differential equations is investigated and found to be a stiff system. A system of stiff differential equations is further discretized by a low-dispersion and low-dissipation implicit Runge–Kutta method and solved by using MATLAB 8.0. The proposed scheme is unconditionally stable. Moreover it is simple, easy to implement and requires less computational time. Finally, the adaptability of the scheme is demonstrated by means of numerical computations by taking three test problems. The present implicit scheme have been compared with existing schemes in literature which shows that the proposed scheme offers more accuracy with less computational time than the numerical schemes given in Jiwari (Comput Phys Comm 183:2413–2423, 2012), Kutluay et al. (J Comput Appl Math 103:251–261, 1998), Kutluay et al. (J Comput Appl Math 167:21–33, 2004).