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Backward difference in time, centered difference in space: \begin{equation} [D_t^- u = D_xD_x u]^n_i \tag{9} \end{equation}
Written out: \begin{equation} \frac{u^{n}_i-u^{n-1}_i}{\Delta t} = \dfc\frac{u^{n}_{i+1} - 2u^n_i + u^n_{i-1}}{\Delta x^2} \tag{10} \end{equation}
Assumption: u^{n-1}_i is computed, but all quantities at the new time level t_n are unknown.
We cannot solve wrt u^n_i because that unknown value is coupled to two other unknown values: u^n_{i-1} and u^n_{i+1} . That is, all the new unknown values are coupled to each other in a linear system of algebraic equations.
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