Taylor series

General Taylor series expansion from calculus: $$f(x+h) = \sum_{i=0}^\infty \frac{1}{i!}\frac{d^if}{dx^i}(x)h^i\tp$$

Here: expand $u^{n-1}$ around $t_n$: $$\begin{align*} u(t_{n-1}) = u(t-\Delta t) &= \sum_{i=0}^\infty \frac{1}{i!}\frac{d^iu}{dt^i}(t_n)(-\Delta t)^i\\ & = u(t_n) - u'(t_n)\Delta t + {\half}u''(t_n)\Delta t^2 + \Oof{\Delta t^3}, \end{align*}$$

• $\Oof{\Delta t^3}$: power-series in $\Delta t$ where the lowest power is $\Delta t^3$
• Small $\Delta t$: $\Delta t \gg \Delta t^3 \gg \Delta t^4$