Lagmental Vicfred

Backward Euler Is A-Stable on the Left Half-Plane by Vicfred

Last updated: Tue 16 April 2024

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

The implicit Euler step damps every test equation with negative real part regardless of step size. This is a compact note, but the quantifiers and hypotheses stay on the page.

Set-up

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

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

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ R(z)=\frac1{1-z},\qquad\Re z\le0\Longrightarrow|R(z)|\le1 $$

The calculation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ y'=-1000y,\ h=0.1:\qquad\begin{cases}R_{\mathrm{FE}}=-99,&\text{explosive},\\R_{\mathrm{BE}}=1/101,&\text{damped}.\end{cases} $$

What survives abstraction

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad y_{n+1}=y_n+h f(t_{n+1},y_{n+1}),\\[5pt] \mathsf{C}\;&:\quad R(z)=\frac1{1-z},\qquad\Re z\le0\Longrightarrow|R(z)|\le1. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] R(z)=\frac1{1-z},\qquad\Re z\le0\Longrightarrow|R(z)|\le1 \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Tue 23 August 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.