  As is well-known, the advantage of the high-order compact difference scheme
(H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$
under the maximum norm. In this article, a new numerical gradient scheme based
on the collocation polynomial and Hermite interpolation is presented. Moreover,
the convergence order of this kind of method is also $O(\tau^2+h^4)$ under the
discrete maximum norm when the space step size is just twice the one of H-OCD
method, which accelerates the computational process and makes the result much
smoother to some extent. In addition, some corresponding analyses are made and
the Richardson extrapolation technique is also considered in time direction.
The results of numerical experiments are also consistent with these theoretical
analysis.
