Abstract: When sixth order WENO schemes are used to solve the nonlinear degenerate parabolic equations, a stencil with more points is needed and the boundary condition is difficult to deal with. To overcome this, a fourth order WENO scheme is proposed. The scheme employs a fourth order WENO reconstruction method to discretize the second order spatial derivative term and a fourth order Runge-Kutta method to advance in time. Compared with the sixth order WENO schemes, the scheme uses a stencil with less points. The linear weights can be any positive numbers with the symmetry requirements and that their sum equals one. These merits make the scheme easy to extend to unstructured meshes. Since no mapping procedure and negative weights are involved, the scheme is simple and efficient. Finally, a number of numerical examples are provided to verify the scheme’s four order accuracy and non-oscillatory property.