Lady Windermere's Fan (mathematics)

In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's play Lady Windermere's Fan, A Play About a Good Woman.

Lady Windermere's Fan for a function of one variable

Let $E(\ \tau ,t_{0},y(t_{0})\ )$ be the exact solution operator so that:

$y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})$

with $t_{0}$ denoting the initial time and $y(t)$ the function to be approximated with a given $y(t_{0})$.

Further let $y_{n}$, $n\in \mathbb {N} ,\ n\leq N$ be the numerical approximation at time $t_{n}$, $t_{0}<t_{n}\leq T=t_{N}$. $y_{n}$ can be attained by means of the approximation operator $\Phi (\ h_{n},t_{n},y(t_{n})\ )$ so that:

$y_{n}=\Phi (\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad$ with $h_{n}=t_{n+1}-t_{n}$

The approximation operator represents the numerical scheme used. For a simple explicit forward euler scheme with step witdth $h$ this would be: $\Phi _{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y(t_{n-1})=(1+h{\frac {d}{dt}})\ y(t_{n-1})$

The local error $d_{n}$ is then given by:

$d_{n}:=D(\ h_{n-1},t_{n-1},y(t_{n-1}\ )\ y_{n-1}:=\left[\Phi (\ h_{n-1},t_{n-1},y(t_{n-1})\ )-E(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\right]\ y_{n-1}$

In abbreviation we write:

$\Phi (h_{n}):=\Phi (\ h_{n},t_{n},y(t_{n})\ )$

$E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )$

$D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )$

Then Lady Windermere's Fan for a function of a single variable $t$ writes as:

$y_{N}-y(t_{N})=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ d_{n}$

with a global error of $y_{N}-y(t_{N})$

Explanation

${\begin{aligned}y_{N}-y(t_{N})&{}=y_{N}-\underbrace {\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})} _{=0}-y(t_{N})\\&{}=y_{N}-\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\underbrace {\sum _{n=0}^{N-1}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})-\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})} _{=\prod _{n=0}^{N-1}\Phi (h_{n})\ y(t_{n})-\sum _{n=N}^{N}\left[\prod _{j=n}^{N-1}\Phi (h_{j})\right]\ y(t_{n})=\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})-y(t_{N})}\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ y_{0}-\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\sum _{n=1}^{N}\ \prod _{j=n-1}^{N-1}\Phi (h_{j})\ y(t_{n-1})-\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\left[\Phi (h_{n-1})-E(h_{n-1})\right]\ y(t_{n-1})\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ d_{n}\end{aligned}}$

See also

• Baker–Campbell–Hausdorff formula
• Numerical error