I think the problem shown in the second image can be solved by the adaptive step size method, while the first can not. The radius still keeps getting larger, only with a lower speed.

waleedlatif1

For anyone confused about the instability that this can cause, a good example that really solidified my understanding of this was the code attached to the most recent discussion. Using that example, we are able to see that regardless of what parameters we change (like increasing the timestep), we are never really able to recover fully from a diverged answer, since each position is computed using the previous values. In an example we did in my discussion, we saw that even increasing delta(t)by a factor of 100x made the rope only temporarily stable, until it diverged and we eventually saw the instability of forward Eulers.

prannaypradeep999

These inaccuracies are because larger step sizes means that Euler's method may overshoot or undershoot the actual solution at a greater rate, leading to oscillations or divergence from the true solution. The instability is related to the concept of numerical stability.

I think the problem shown in the second image can be solved by the adaptive step size method, while the first can not. The radius still keeps getting larger, only with a lower speed.

For anyone confused about the instability that this can cause, a good example that really solidified my understanding of this was the code attached to the most recent discussion. Using that example, we are able to see that regardless of what parameters we change (like increasing the timestep), we are never really able to recover fully from a diverged answer, since each position is computed using the previous values. In an example we did in my discussion, we saw that even increasing delta(t)by a factor of 100x made the rope only temporarily stable, until it diverged and we eventually saw the instability of forward Eulers.

These inaccuracies are because larger step sizes means that Euler's method may overshoot or undershoot the actual solution at a greater rate, leading to oscillations or divergence from the true solution. The instability is related to the concept of numerical stability.