$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} $$

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Simplest method: run a few algorithmic steps by hand

Use a calculator (\( I=0.1 \), \( \theta=0.8 \), \( \Delta t =0.8 \)): $$ A\equiv \frac{1 - (1-\theta) a\Delta t}{1 + \theta a \Delta t} = 0.298245614035$$ $$ \begin{align*} u^1 &= AI=0.0298245614035,\\ u^2 &= Au^1= 0.00889504462912,\\ u^3 &=Au^2= 0.00265290804728 \end{align*} $$

See the function verify_three_steps in decay_verf1.py.

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