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The ODE problem has a continuous and discrete version
Continuous problem
\begin{equation}
u' = -au,\ t\in (0,T], \quad u(0)=I
\tag{1}
\end{equation}
(varies with a continuous t )
Discrete problem
Numerical methods applied to the continuous problem turns it into
a discrete problem
\begin{equation}
u^{n+1} = \mbox{const} u^n, \quad n=0,1,\ldots N_t-1, \quad u^n=I
\tag{2}
\end{equation}
(varies with discrete mesh points t_n )
