Order of Convergence

Order of convergence describes how quickly a sequence of numerical approximations approaches the exact solution as the step size $dt\to 0$. If the error $E(dt)$ satisfies

$$E(dt)=O(d{t}^p)$$

then the method has order of convergence $p$.

This is closely related to the Order of a method, which is defined in terms of the Global truncation error. A method of order $p$ converges at rate $p$: halving $dt$ reduces the error by a factor of $2^p$.

Convergence requires both consistency and stability (see the Lax Equivalence Theorem).

Order of a method | Global truncation error | Convergence | Lax Equivalence Theorem