 # 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. kominis3q 2022-07-23 Answered
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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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 $\mu =2\cdot {10}^{-16}$ (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".