We propose a characteristic centered difference method for a class of nonlinear evolution equations on nonregular grids.Approximate solution and error estimate of u,u_t,u_x,u_tx are obtained.The compuational load of the method is the same as those of the characteristic difference method based on linear interpolation.And the error order of the approximate solution is the same as one of the characteristic difference method based on quadratic interpolation.Moreover,the first derivative of u,u_t in space shows super-covergent order error estimate.Numerical results demonstrate feasibility and efficiency of the methods.