In particular, if Euler's method is implemented on a computer, what's the minimum step size that can be used before rounding errors cause the Euler approximations to become completely unreliable?

I presume it's when step size reaches the machine epsilon? E.g. if machine epsilon is e-16, then once step size is roughly e-16, the Euler approximations are unreliable.

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    $\begingroup$ You may get better answers at the scientific computing stack exchange $\endgroup$ – Nick Alger Apr 14 '19 at 22:39

It depends on the problem. Basically, you have two sources of errors in each step, the rounding errors that are some multiple of the machine constant $\mu=2\cdot10^{-16}$ (scaled by the scale of the functions involved) and the local truncation error, which depends on derivatives of the ODE function (thus also involving the function scale) and the square $h^2$ of the step size. You want the truncation error to dominate the rounding errors, which in the simplest case demands $h^2>\mu$ or $h>\sqrt\mu\sim 10^{-8}$. This changes if the ODE function is in some sense "strongly curved".

  • $\begingroup$ What is $\mu$ ? $\endgroup$ – DataBSc Apr 14 '19 at 22:57
  • $\begingroup$ It is the machine constant of the floating point type, it is now introduced at the first mention of it. $\endgroup$ – Lutz Lehmann Apr 14 '19 at 23:01

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