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 1892 play Lady Windermere's Fan, A Play About a Good Woman.
(
τ ,
t
0
t
0
)
{\displaystyle E(\ \tau ,t_{0},y(t_{0})\ )}
be the exact solution operator so that: with
0
denoting the initial time and
y ( t )
the function to be approximated with a given
y (
0
{\displaystyle y(t_{0})}
.
be the numerical approximation at time
can be attained by means of the approximation operator
so that: The approximation operator represents the numerical scheme used.
For a simple explicit forward Euler method with step width
Euler
{\displaystyle \Phi _{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y_{n-1}=(1+h{\frac {d}{dt}})\ y_{n-1}}
The local error
is then given by: In abbreviation we write: Then Lady Windermere's Fan for a function of a single variable
writes as:
with a global error of
{\displaystyle {\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 _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})-\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}}}