Convergence

A numerical algorithm converges if the numerical solution approaches the exact solution as the step size $dt\to 0$.

Formally, a method converges if

$$\underset{i}{\max }|y_i-y(t_i)|\to 0\text{as }dt\to 0$$

The rate at which this occurs is the Order of convergence.

Consistency

A method is consistent if the exact solution satisfies the numerical scheme as $dt\to 0$. Informally, the discretisation must become exact in the limit of infinitely fine resolution.

Relationship to Stability

Convergence, consistency, and stability are linked by the Lax Equivalence Theorem:

$$\text{consistent}+\text{stable}⟺\text{convergent}$$

This is useful because consistency and stability are often easier to check individually than convergence directly. Proving stability via the amplification factor and checking that the Local truncation error vanishes as $dt\to 0$ (consistency) together guarantee convergence.

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