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Compare computed numerical solution with a closed-form exact discrete solution (if possible).
Define $$ A = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a \Delta t}$$ Repeated use of the \( \theta \)-rule: $$ \begin{align*} u^0 &= I,\\ u^1 &= Au^0 = AI\\ u^n &= A^nu^{n-1} = A^nI \end{align*} $$
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