In our foregoing paper[8] we had designed a nonlinear conservative difference scheme of second-order Godunov type with piecewise-linear reconstruction,in which the slope of the reconstructed function in each grid cell can be computed by dissipating the entropy. Such a scheme satisfies the entropy condition,and computes not only numerical solution but also numerical entropy; thus, it is different from all former conservative schemes. A socalled entropy dissipator in the scheme, which dissipates the entropy in each grid cell in the computation, plays an important role in stabilizing the computation. The entropy dissipator designed in is quite complicated. In this paper, we numerically discuss why entropy must be dissipated and how much entropy should be dissipated. A new entropy dissipator, based on the second-order difference of the numerical solution is given. Numerical examples are presented to show how the entropy dissipator suppresses nonphysical oscillations near discontinuities.