# 6.3. A Global Error Bound for One Step Methods#

**References:**

Subection 6.2.1

*Local and global truncation error*in [Sauer, 2022].Section 5.2

*Euler’s Method*in [Burden*et al.*, 2016].Section 8.5 of [Kincaid and Chenney, 1990]

All the methods seen so far for solving ODE IVP’s are *one-step methods*:
they fit the general form

For example, Euler’s Method has

the Explicit Midpoint Method (Modified Euler) has

and even the Runge-Kutta method has a similar form, but it is long and ugly.

For these, there is a general result that gives a bound on the globl truncation error (“GTE”) once one has a suitable bound on the local truncation error (“LTE”). This is very useful, because bounds on the LTE are usually far easier to derive.

When solving the ODE IVP

on interval \(t \in [a, b]\) by a one step method, one has a bound on the local truncation error

and the ODE itself satisfies the *Lipschitz Condition* that for some constant \(K\),

then there is a bound on the global truncation error:

So yet again, there is a loss of one factor of \(h\) in going from local to global error, as first seen with the composite rules for definite integrals.

We saw a glimpse of this for Euler’s method, in the section Basic Concepts and Euler’s Method where the Taylor’s Theorem error formula canbe used to get the LTE bound

and this leads to to GTE bound

## 6.3.1. Order of accuracy for the basic Runge-Kutta type methods#

For Euler’s method, it was stated in section Basic Concepts and Euler’s Method, (and verified for the test case of \(du/dt = ku\)) that the global truncation error is of first order n step-size \(h\):

The Explicit (and Implicit) Trapezoid and Midpoint rules, the local truncation error is \(O(h^3)\) and so their global truncation error is \(O(h^2)\) — they are second order accurate, just as for the corresponding approximate integration rules.

The classical Runge-Kutta method, has local truncation error \(O(h^5)\) and so its global truncation error is \(O(h^4)\) — just as for the composite Simpson’s Rule, to which it corresponds for the “integration” case \(dy/dt = f(t)\).