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The \( \theta \)-rule: $$ [\bar D_t u = -a\overline{u}^{t,\theta}]^{n+\theta} \tp $$
Truncation error: $$ [\bar D_t \uex + a\overline{\uex}^{t,\theta} = R]^{n+\theta} \tp $$ Use (11)-(12) and (17)-(18) along with \( \uex'(t_{n+\theta}) + a\uex(t_{n+\theta})=0 \) to show
$$ \begin{align} R^{n+\theta} = &({\half}-\theta)\uex''(t_{n+\theta})\Delta t + \half\theta (1-\theta)\uex''(t_{n+\theta})\Delta t^2 + \nonumber\\ & \half(\theta^2 -\theta + 3)\uex'''(t_{n+\theta})\Delta t^2 + \Oof{\Delta t^3} \end{align} $$ Note: 2nd-order scheme if and only if \( \theta =1/2 \).
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