Numerical Simulation of Burgers’ Equation

Abstract

An exponential finite difference technique is first presented by Bhattacharya for one‐
dimensional unsteady state. In this study, the exponential finite difference technique was
used to solve the Burgers’ equation in one‐dimensional with different value of h (step
size). Burgers’ equation is considered in this study because the equation governing simple
nonlinear diffusion process. Since the Burgers’ equation is nonlinear, the Hopf‐Cole
transformation is applied to the linear heat equation which was converted from Burgers’
equation. Then, the exponential finite difference methods are used to obtain numerical
solution. Three techniques have been implemented namely explicit exponential finite
difference method, implicit exponential finite difference method and modified Burgers’
equation using explicit exponential finite difference method. In the solution process, the
explicit exponential finite difference method used a direct to solve the Burgers’ equation
while the implicit exponential finite difference method leads to a system of nonlinear
equation. At each time‐level, Newton’s method is used to solve the nonlinear system. The
solution of the one‐dimensional modified Burgers’ equation is using the explicit
exponential finite difference method. The solution process has discretized the time
derivative and spatial derivative using exponential finite difference technique. Numerical
solutions for each method are compared with exact solution and the results obtained
using the three methods are precise and reliable. The percent errors are computed and
found to be sufficiently small.