Lagmental Vicfred

Forward Euler Uses One Tangent Step by Vicfred

Last updated: Fri 15 May 2020

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

The first-order Taylor approximation advances with the vector field at the start of the step. The point is to make the formal expression readable enough to audit line by line.

The data

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 \(y_n\approx y(t_n)\).

$$ y_{n+1}=y_n+h f(t_n,y_n) $$

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.

$$ y(t_{n+1})-y(t_n)-hf(t_n,y(t_n))=O(h^2) $$

Derivation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ y'=\lambda y\Longrightarrow y_n=(1+h\lambda)^ny_0,\qquad y(t_n)=e^{\lambda nh}y_0 $$

Invariant content

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad y_{n+1}=y_n+h f(t_n,y_n),\\[5pt] \mathsf{C}\;&:\quad y(t_{n+1})-y(t_n)-hf(t_n,y(t_n))=O(h^2). \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] y(t_{n+1})-y(t_n)-hf(t_n,y(t_n))=O(h^2) \end{gathered}} $$

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 Fri 25 August 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.