Implementation

Model: $$u'(t) = -au(t),\quad t\in (0,T], \quad u(0)=I$$

Numerical method: $$u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n$$ for $\theta\in [0,1]$. Note

• $\theta=0$ gives Forward Euler
• $\theta=1$ gives Backward Euler
• $\theta=1/2$ gives Crank-Nicolson