---
tag: T220HF0562
kind: concept
name: Local truncation error
msc: [65L70, 65L05, 65L06]
specializes: [4AKCQERM89]
sources:
  - url: https://en.wikipedia.org/wiki/Truncation_error_(numerical_integration)
---

## Summary

The local truncation error of a method for an [[initial value problem|0FXDW992DS]] is the error of one step started from exact values of the solution.
It measures how well the formula of the method matches the differential equation, and it defines the order of consistency of the method.
Together with a stability condition it controls the global error.
For a one-step method with local error $O(h^{p+1})$ per step, the global error after $O(1/h)$ steps is $O(h^p)$.
It is a special case of [[truncation error|4AKCQERM89]].

## Explanations

### L1 human

Imagine standing exactly on the true solution curve and taking one step with the method.
The local truncation error is how far from the curve that single step lands.
For the [[Euler method|0JARTXX4XP]] applied to $y' = y$ starting at $y = 1$, one step of length $h$ gives $1 + h$, while the truth is $e^h = 1 + h + \frac{h^2}{2} + \cdots$, so the local error is about $\frac{h^2}{2}$.
Over the whole interval many such small errors add up to the global error.