In this paper, an implicit finite difference method based on the Crank–Nicolson method is proposed for the numerical solution of the one-dimensional Burger–Fisher equation. The Crank–Nicolson scheme provides a system of nonlinear difference equations, which is solved by an integration of the Jacobian-Free-Newton-Krylov (JFNK) and GMRES methods. Various numerical examples are given to demonstrate the efficiency of the proposed scheme. Comparison of the computed solutions with the analytical ones demonstrates the accuracy of this proposed method.