On Schemes for Exponential Decay

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.