Lagmental Vicfred

Absolute Stability Is Tested on y Prime Equals lambda y by Vicfred

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

A one-step method is stable at z when its amplification factor has modulus at most one. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

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

$$ y'=\lambda y,\qquad z=h\lambda $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ y_{n+1}=R(z)y_n,\qquad\mathcal S=\{z\in\mathbf C:|R(z)|\le1\} $$

Worked algebra

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \begin{array}{c|c|c}\text{method}&R(z)&\mathcal S\cap\mathbf R_{<0}\\\hline\text{forward Euler}&1+z&[-2,0]\\\text{backward Euler}&(1-z)^{-1}&(-\infty,0]\end{array} $$

Conceptual compression

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad y'=\lambda y,\qquad z=h\lambda,\\[5pt] \mathsf{C}\;&:\quad y_{n+1}=R(z)y_n,\qquad\mathcal S=\{z\in\mathbf C:|R(z)|\le1\}. \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] y_{n+1}=R(z)y_n,\qquad\mathcal S=\{z\in\mathbf C:|R(z)|\le1\} \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Wed 18 February 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.