Numerical solution method
- 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 \( t_n \)
Numerical scheme:
$$
u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n,
\quad n=0,1,\ldots,N-1
$$