 Consistency of a numerical method

For a one-step approximation $y_{n+1}=y_n+h\varphi (t_n,y_n,h)$ to an ordinary differential equation, insert the exact solution and define the unscaled local truncation errorConsistency of a numerical method requires ${\tau }_{n+1}/h\to 0$ uniformly on the finite interval being approximated, under the stated smoothness assumptions. Merely asking ${\tau }_{n+1}\to 0$ is too weak: the useless update $y_{n+1}=y_n$ has defect $h{y}^{\prime }(t_n)+O(h^2)$, which tends to zero but generally fails the scaled test. Here $O$ denotes Big O notation. MIT uses this division-by-step convention.^[ref]

For the Euler method, Taylor's theorem gives ${\tau }_{n+1}=h^2{y}^{\prime \prime }({\xi }_n)/2$. A bounded second derivative therefore gives local truncation error $O(h^2)$ and a scaled defect $O(h)$. Under a uniform Lipschitz continuity assumption on $f$ in $y$, within a region containing the exact and numerical solutions, numerical stability controls accumulation over about $T/h$ steps. This yields global error $O(h)$ with exact starting data. Consistency of a numerical method checks each step; numerical stability controls their combination.