Averaging in time is necessary in the Crank-Nicolson scheme

Right-hand side term: $$\frac{1}{\Delta x^2}\left(u^{n+\half}_{i-1} - 2u^{n+\half}_i + u^{n+\half}_{i+1}\right)$$

Problem: $u^{n+\half}_i$ is not one of the unknowns we compute.

Solution: replace $u^{n+\half}_i$ by an arithmetic average: $$u^{n+\half}_i\approx \half\left(u^{n}_i +u^{n+1}_{i}\right)$$

In compact notation (arithmetic average in time $\overline{u}^t$): $$[D_t u = \dfc D_xD_x \overline{u}^t]^{n+\half}_i$$