= Consistency of a numerical method
= Numerical consistency
{synonym}
For a one-step approximation $y_{n+1}=y_n+h\phi(t_n,y_n,h)$ to an <ordinary differential equation>, insert the exact solution and define the unscaled <local truncation error>
$$
\tau_{n+1}=y(t_n+h)-y(t_n)-h\phi(t_n,y(t_n),h).
$$
<Consistency of a numerical method> requires $\tau_{n+1}/h\to0$ uniformly on the finite interval being approximated, under the stated smoothness assumptions. Merely asking $\tau_{n+1}\to0$ is too weak: the useless update $y_{n+1}=y_n$ has defect $h y'(t_n)+O(h^2)$, which tends to zero but generally fails the scaled test. Here $O$ denotes <Big O notation>. MIT uses this division-by-step convention.https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/a9d2bd9be098f0ada172af40379a17cc_MIT18_330S12_Chapter5.pdf[Laurent Demanet, MIT 18.330, Chapter 5, Definition 6, 2012]{ref}
For the <Euler method>, <Taylor's theorem> gives $\tau_{n+1}=h^2y''(\xi_n)/2$. 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.
Back to article page
