Lagmental Vicfred

Local Truncation Error Accumulates into Global Error by Vicfred

Last updated: Mon 22 August 2011

tags: mathematics, ordinary differential equations, Runge--Kutta methods, stability

A one-step method with local error order p plus one typically has global error order p under stability. A small computation will anchor the general statement before the abstraction takes over.

Notation

An initial-value problem \(y'=f(t,y)\), \(y(t_0)=y_0\) generates a flow when existence and uniqueness hold. A numerical method advances discrete states \(y_n\approx y(t_n)\).

$$ \tau_{n+1}=\frac{y(t_{n+1})-\Phi_h(t_n,y(t_n))}{h} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \tau_{n+1}=O(h^p)\Longrightarrow\max_{0\le n\le T/h}|y(t_n)-y_n|=O(h^p) $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ e_{n+1}\le(1+hL)e_n+Ch^{p+1}\Longrightarrow e_n\le\frac C L\bigl(e^{LT}-1\bigr)h^p $$

Interpretation

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad \tau_{n+1}=\frac{y(t_{n+1})-\Phi_h(t_n,y(t_n))}{h},\\[5pt] \mathsf{C}\;&:\quad \tau_{n+1}=O(h^p)\Longrightarrow\max_{0\le n\le T/h}|y(t_n)-y_n|=O(h^p). \end{aligned} $$

Limit of the argument

Order of accuracy and stability region are separate. A high-order explicit method can fail spectacularly on a stiff equation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \tau_{n+1}=O(h^p)\Longrightarrow\max_{0\le n\le T/h}|y(t_n)-y_n|=O(h^p) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Thu 23 September 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.