Step 3: Replacing derivatives by finite differences

Use a forward difference in time and a centered difference in space (Forward Euler scheme): $$\begin{equation} [D_t^+ u = \dfc D_xD_x u]^n_i \tag{6} \end{equation}$$

Written out, $$\begin{equation} \frac{u^{n+1}_i-u^n_i}{\Delta t} = \dfc \frac{u^{n}_{i+1} - 2u^n_i + u^n_{i-1}}{\Delta x^2} \tag{7} \end{equation}$$

Initial condition: $u^0_i = I(x_i)$, $i=0,1,\ldots,N_x$.