2.2.2 Simple ODE: examples

**Figure 1:** First order integration of ODE with different time-steps.
$\includegraphics[width=8.0cm]{images/pde1d_ode_smalldt.ps}$ $\includegraphics[width=8.0cm]{images/pde1d_ode_bigdt.ps}$

Figure 1 shows examples for Eq. (75) for $\bgroup\color{DEFcolor}$a=1$\egroup$ . The

$\displaystyle \mbox{{\color{HIGH1color}criterion for stability}:\index{stability!of ODE}}$	$\textstyle \hspace{2cm}$	$\displaystyle \Delta t< 2/a$	(76)
$\displaystyle \mbox{{\color{HIGH1color}criterion for positivity}:}$	$\textstyle \hspace{2cm}$	$\displaystyle \Delta t< 1/a$	(77)

have a simple interpretation: Too large time-steps lead to conflicts with the actual curvature of the solution.