Study guide: Finite difference methods for wave motion

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Discretizing the variable coefficient (3)

These intermediate results are now combined to $$ \begin{equation} \left[ \frac{\partial}{\partial x}\left( q(x) \frac{\partial u}{\partial x}\right)\right]^n_i \approx \frac{1}{\Delta x^2} \left( q_{i+\half} \left({u^n_{i+1} - u^n_{i}}\right) - q_{i-\half} \left({u^n_{i} - u^n_{i-1}}\right)\right) \tag{24} \end{equation} $$

In operator notation: $$ \begin{equation} \left[ \frac{\partial}{\partial x}\left( q(x) \frac{\partial u}{\partial x}\right)\right]^n_i \approx [D_xq D_x u]^n_i \tag{25} \end{equation} $$

Remark.

Many are tempted to use the chain rule on the term $ \frac{\partial}{\partial x}\left( q(x) \frac{\partial u}{\partial x}\right) $, but this is not a good idea!