Consistency

A numerical method is consistent if the local truncation error per step vanishes as the step size $dt\to 0$:

$$\frac{\text{LTE}}{dt}\to 0\text{as }dt\to 0$$

Equivalently, the discrete scheme must recover the correct differential equation in the limit of infinitely fine resolution.

Consistency is a necessary condition for convergence but is not sufficient on its own. The Lax Equivalence Theorem states that for a well-posed problem, consistency combined with stability is both necessary and sufficient for Convergence.

All standard Runge-Kutta methods (explicit Euler, midpoint, Ralston, RK4, implicit trapezoid) are consistent by construction - their Local truncation error is at least $O(d{t}^2)$, so the ratio LTE$/dt$ is at least $O(dt)\to 0$. The question of whether a method converges therefore reduces to checking its stability.

Local truncation error | Lax Equivalence Theorem | Stability of a method | Convergence | Order of a method