# What's the minimum step size that can be used in Euler's method before it becomes unreliable?

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.

• You may get better answers at the scientific computing stack exchange 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".
• What is $\mu$ ?