# Global Error Bounds for One Step Methods — Preliminary

## Contents

# 31. Global Error Bounds for One Step Methods — Preliminary#

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.

**Theorem**

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 Solving Initial Value Problems for Ordinary Differential Equations, Part 1: Basic Concepts and Euler’s Method — Python version, where the Taylor’s Theorem error formula canbe used to get the LTE bound

and this leads to to GTE bound

## 31.1. Order of accuracy for the basic Runge-Kutta type mehods#

For Euler’s method, it was stated in section Solving Initial Value Problems for Ordinary Differential Equations, Part 1: Basic Concepts and Euler’s Method — Python version, (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.

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