Practice question · Multiple choice
Euler's method for a differential equation takes a step along the current tangent. Why does the error accumulate over many steps rather than staying at the per-step level?
Hints
- After the first step you are on a slightly wrong solution curve. What does the second step follow?
- Local error O(h²) per step, and how many steps to cross a fixed interval?
Show the answer
A. Because each step starts from a point already slightly wrong.
Why
The first step lands slightly off, and the second follows the tangent to whichever curve passes through that point, building on the error rather than correcting it. Local O(h²) over 1/h steps gives global O(h), so halving the step only halves the error. RK4 evaluates four times per step for O(h⁴), a far better return, which is why it is the workhorse.
Practise Simulation and Monte Carlo Methods
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