Results

The picture to the right shows how the error in the iterated solution behaves as a function of time and iteration step. Since the initial conditions are known, the error at the first time-step will be zero at every iteration. Trapezoidal-rule uses this converged initial-point to converge the next time-step. Then once the second time-step starts to converge, the third starts to converge, and so on. At any given iteration, we can see which points have converged, which seem to be converging, and which are behaving chaoticly. A couple questions arise when one looks at the graph.

Should we spend time iterating in the chaotic region, if the points are not able to converge? Or should we create a "time-window". Using the latest time-step to converge as the begining of the window, and n-steps later as the end of the window. Is there a way to predict how long to make this window? Or even better, can we predict where the solution is converging at any given step? In order to define an optimal time-window, we should answer this last question.

How can we approximate where the iterated solution is surely approaching to true solution. Or, rather, where is the error surely converging to zero?

The major result of this work is the near-linear relationship between iteration number and t. Where t is the end of the time region where we surely expect convergence. In other words, the estimate for the error must be converging in the interval [0,t] at iteration k, with t a near-linear function of k. The line across the above graph shows this relationship. While the line does not follow the graph exactly, it is remarkablely close. t is directly proportional to the kth-root of k-factorial. The constant of proportionality is only dependent on the initial conditions. What is remarkable is that the kth-root of k-factorial has a linear-correlation coefficient of .999 for k=1 to 60. This is extremely linear.