The ODE problem is solved by a finite difference scheme

• Mesh in time: $0= t_0 < t_1 \cdots < t_N=T$
• Assume constant $\Delta t = t_{n}-t_{n-1}$
• $u^n$: numerical approx to the exact solution at $t_n$

The $\theta$ rule, $$u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n, \quad n=0,1,\ldots,N-1$$ contains the Forward Euler ($\theta=0$), the Backward Euler ($\theta=1$), and the Crank-Nicolson ($\theta=0.5$) schemes.