Scientific Computing - Quiz 1 - Question 4

Problem:

Using a Taylor expansion for $y(t+\Delta t) = y_{n+1}$ and $y(t-\Delta t)=y_{n-1}$, derive an Euler-type formula for the future state of the system $y' = f(y,t)$.

Solution:

The Taylor expansions are:

$y_{n+1} = y(t + \Delta t) = y(t) + \Delta t \frac{dy}{dt}(t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_1) = y(t) + \Delta t f(y, t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_1)$
$y_{n-1} = y(t - \Delta t) = y(t) - \Delta t \frac{dy}{dt}(t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_2) = y(t) - \Delta t f(y, t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_2)$

Now subtract the equations to get our answer
$\begin{eqnarray*} y_{n+1} - y_{n-1} &=& (y(t) + \Delta t f(y, t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_1)) - (y(t) - \Delta t f(y, t) + \frac{\Delta t^2}{2} \frac{d^2y}{dt^2}(c_2)) \\ &=& 2\Delta t f(y, t) + \frac{\Delta t^2}{2}(\frac{d^2y}{dt^2}(c_1) -\frac{d^2y}{dt^2}(c_2)) \\ y_{n+1} &=& y_{n-1} + 2\Delta t f(y, t) + \frac{\Delta t^2}{2}(\frac{d^2y}{dt^2}(c_1) -\frac{d^2y}{dt^2}(c_2)) \\ & \approx & y_{n-1} + 2\Delta t f(y, t) \end{eqnarray*}$