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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 \( \overline{u}^t \)): $$ [D_t u = \dfc D_xD_x \overline{u}^t]^{n+\half}_i$$
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