Integrating "Beyond Infinity" and Back
Consider the simple ordinary differential equation with initial condition . The solution is obviously given by . It tends to infinity as tends to 1. We now consider a time-discrete approximation to this solution as provided by the asynchronous leapfrog method. Since the evolution step formula does not involve potentially undefined operations, any such discrete approximation is well defined for all its values. Of course, the approximated values grow dramatically and will soon transcend what can be represented even with Mathematica's arbitrary-precision numbers. Since the asynchronous leapfrog method is a reversible integration method, we should be able to go along each finite discrete trajectory back to its initial point. If the final point was "close to infinity", the computation needs to be done with a large number of digits in order to come back to its initial point. This is what the present Demonstration studies. Values are input by means of setter bars instead of sliders since updating the curve for higher precision and may take a few seconds. (This does not affect Autorun, which uses machine precision throughout.) The reversed trajectory is marked by red dots and one easily sees (if the box 'show also the reversed trajectory' is activated) whether the reversed trajectory reaches the initial point. In all cases where it fails to do so, increasing the value of precision will finally solve the problem. In some cases, precision up to 2000 is needed.[more]
As the integration method is set up here, it creates a trajectory that always grows, even if we have crossed the singularity of the exact solution. Referring to this situation as going beyond infinity, as I did in the title, is, of course, not to be taken literally. A slight modification of the asynchronous leapfrog method can allow the method to "integrate across the singularity" and to approximate the exact solution well also for . This will be analyzed in a future Demonstration.[less]
Snapshot 1: Maximum stress, reaching the largest numbers. The integration is from to with ; precision 1000 turns out to be insufficient for reversibility.
Snapshot 2: Reversible trajectory that gets lost in machine precision. Here the integration is from to with ; due to the larger value of , there are fewer steps in the critical domain and the values are less gigantic; so a precision of 250 is sufficient for getting a reversible trajectory.
Snapshot 3: Using machine precision, together with the remaining settings of Snapshot 2, spoils reversibility.
 Ulrich Mutze An Asynchronous Leap-Frog Method, 2008., http://www.ma.utexas.edu/mp_arc/c/08/08-197.pdf.
 Alternate link for : http://www.ulrichmutze.de/articles/leapfrog4.pdf.