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 1892 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 $$y_n$$, $$n \in \N,\ n\le N$$ be the numerical approximation at time $$t_n$$, $$t_0 < t_n \le T = t_N$$. $$y_n$$ 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 method with step width $$h$$ this would be: $$\Phi_{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y_{n-1} = (1 + h \frac{d}{dt})\ y_{n-1}$$

The local error $$d_n$$ 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
$$\begin{align} y_N - y(t_N) &{}= y_N - \underbrace{\prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) + \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0)}_{=0} - y(t_N) \\ &{}= y_N - \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) + \underbrace{\sum_{n=0}^{N-1}\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n) - \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n)}_{=\prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0)-\sum_{n=N}^{N}\left[\prod_{j=n}^{N-1} \Phi(h_j)\right]\ y(t_n) = \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) - y(t_N) } \\ &{}= \prod_{j=0}^{N-1}\Phi(h_j)\ y_0 - \prod_{j=0}^{N-1}\Phi(h_j)\ y(t_0) + \sum_{n=1}^N\ \prod_{j=n-1}^{N-1} \Phi(h_j)\ y(t_{n-1}) - \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n) \\ &{}= \prod_{j=0}^{N-1}\Phi(h_j)\ (y_0-y(t_0)) + \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j) \left[ \Phi(h_{n-1}) - E(h_{n-1}) \right] \ y(t_{n-1}) \\ &{}= \prod_{j=0}^{N-1}\Phi(h_j)\ (y_0-y(t_0)) + \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ d_n \end{align}$$