In practice, you would proceed as follows. If the curves converge then the solution is to all appearances stable, and your accuracy is sufficient. If employing a constant stepsize, perform a similar analysis by successively halving the stepsize. The following example is one that you may wish to experiment with. The convergence, showing stability, should be quite illuminating.
Many systems have invariant quantities.
In general, knowledge about the qualitative behavior of any dependent variable may be used to check the quality of the solution. A rough idea of how error is propagated is obtained by viewing a family of solution curves about the numerical solution in question, obtained by varying the initial conditions. On the other hand, if the curves do not diverge sharply then any error that is present will in all likelihood not increase by more than an order of magnitude or so over the interval. Problems exhibiting no sharp divergence of neighboring solution curves are sometimes called well-conditioned . Go to the first , previous , next , last section, table of contents .