|
|
|
Idea 1: let the ODE hold at \( t_{n+\half} \) $$ u'(t_{n+\half}) = -au(t_{n+\half})$$
Idea 2: approximate \( u'(t_{n+\half} \) by a centered difference $$ \begin{equation} u'(t_{n+\half}) \approx \frac{u^{n+1}-u^n}{t_{n+1}-t_n} \tag{10} \end{equation} $$
Problem: \( u(t_{n+\half}) \) is not defined, only \( u^n=u(t_n) \) and \( u^{n+1}=u(t_{n+1}) \)
Solution: $$ u(t_{n+\half}) \approx \half(u^n + u^{n+1}) $$
|
|
|