Efficient and Accurate Finite Difference Schemes for Solving One-dimensional Burgers’ Equation

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].