Round-Off Error
Round-off error arises from the finite precision of floating-point arithmetic. Real numbers are stored with a fixed number of significant digits (e.g. 64-bit IEEE double precision gives roughly 15-16 decimal digits), so any number that cannot be represented exactly is rounded to the nearest representable value.
Unlike Local truncation error - which decreases as the step size shrinks - round-off error accumulates with more arithmetic operations. For very small , the truncation error may decrease but the number of steps increases, causing accumulated round-off to eventually dominate.
In practice, this means there is an optimal step size: too large gives large truncation error, too small gives large accumulated round-off. For the methods in this module, round-off is not usually the binding constraint; the stability condition (see FTCS scheme) imposes a much tighter upper limit on than round-off imposes a lower limit.
Local truncation error | Global truncation error | Absolute error | Order of a method