Taking More Steps
One big leap is inaccurate. So instead of jumping from t = 1 to t = 5 in a single step, we chop the journey into many small steps and take them one after another.
We just apply the same rule over and over, feeding each answer into the next step. Written as a repeating (discrete) equation:
p[k+1] = p[k] + Δt·(a·p[k]) for k = 0, 1, 2, …
p[k] is the current estimate; p[k+1] is the next one. Same 'Future = current + step × dynamics', looped.
Slide Δt to change how many steps you take. Fewer big steps wander off; more small steps hug the true curve:
Each green dot is one Euler step of size Δt. Fewer, bigger steps drift far from the grey exact curve; more, smaller steps hug it. Notice the error grows at every step — that build-up is the global error, so predicting further into the future is always harder.
Two errors, one lesson. Each step adds a little local error, and those add up step after step into a growing global error. That's why the estimate drifts further from the truth the longer you run — predicting far into the future is always harder. Smaller Δt shrinks the drift, but costs more steps (computation). Always a trade-off.