A numerical method often produces a family of approximations $u_h$ controlled by a refinement parameter $h$, such as a step size or mesh spacing.

The method converges when

$$u_h\to u$$

as $h\to0$, where $u$ is the exact quantity being approximated.

A method has order $p$ when, for sufficiently small $h$, its leading error behaves approximately like

$$|u_h-u|\approx C h^p$$

for some constant $C$. Halving $h$ should then reduce that leading error by roughly a factor of $2^p$ once the asymptotic regime is reached.

Convergence is distinct from obtaining a plausible-looking answer at one resolution. Refinement studies compare successive approximations to check whether the result is approaching a stable limit at the expected rate.

At very fine resolution, roundoff or other error sources can eventually dominate, so smaller $h$ does not guarantee indefinitely improving accuracy.