Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation

Abstract

In this paper, two efficient fourth-order compact finite difference algorithms have been developed to solve the one-dimensional Burgers’ equation: u _t +u u _x =ε u _xx . The methods are based on the Hopf–Cole transformation, Richardson's extrapolation, and multilevel grids. In both methods, we first transform the original nonlinear Burgers’ equation into a linear heat equation: w _t =ε w _xx using the Hopf–Cole transformation, which is given as u=−2ε (w _x /w). In the first method, the resulted heat equation is solved by the second-order accurate Crank–Nicholson algorithm while w _x is approximated by central finite difference, which is also second-order accurate. Richardson's extrapolation technique is then applied in both time and space to obtain fourth-order accuracy. In the second method, to reduce the cancellation error in approximating w _x , we derive the heat equation satisfied by w _x , which is then solved by the Crank–Nicholson algorithm. The original Dirichlet boundary condition is transformed into the Robin boundary condition, which is also approximated using second-order central finite difference. Finally, Richardson's extrapolation and multilevel grid techniques are applied in both time and space to obtain fourth-order accuracy. To study the efficiency, accuracy and robustness, we solved two numerical examples and the results are compared with those of two other higher-order methods proposed in W. Liao [An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206(2) (2008), pp. 755–764] and I.A. Hassanien, A.A. Salama, and H.A. Hosham [Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput. 170 (2005), pp. 781–800].

Keywords:

• Burgers’ equation
• Multilevel grids
• Richardson's extrapolation
• finite difference scheme
• higher-order compact algorithm

2000 AMS Subject Classifications :

• 65M06
• 65M55
• 65L12
• 65N06

Acknowledgements

The work of the first author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank the anonymous referees for their efforts and constructive comments on the revision of the manuscript.