An Improvement?

Consider the operation:

$$y_n \mapsto {\mathchoice{{\textstyle{\frac14}}}{{\textstyle... ...riptscriptstyle{1/4}}}}\left(y_{n+1} + 2 y_n + y_{n-1}\right)$$ (2.60)

By applying the von Neumann trick you should be able to deduce the effect of such an operation. Now try applying the transformation (2.60) to to $x$ and $u$ after a few iterations. How does the behaviour of your simulation change?