Improved Euler - Derivation

Consider a Taylor series expansion of the solution of the differential equation. Let the equation be

$$\frac{ dy}{d t } = f( y, t ).$$

The unknown function value at $t_0+h$ is obtained via an expansion of the solution about $t=t_0$, yielding

$$y(t_0+h) = y(t_0) + h \frac{ d y}{d t }(t_0) + \frac{h^2}{2} \frac{ d y^2}{d t^2 }(t_0) +O(h^3).$$

In this equation $h=dt$ is the time step.

The Predictor-Corrector Improved Euler method takes into account of the next term in the series. This requires knowing $\frac{ d y^2}{d t^2 }(t_0)$. For simplicity in the derivation, let us assume that $f(y,t)$ depends only on $y(t)$. Then

$$\frac{ d y^2}{d t^2 } = \frac{ d f}{d t } = \frac{ \parti... ...ac{ d y}{d t } = \frac{ \partial f}{\partial y } f(y) = f' f.$$

$f'$ can be approximated in terms of a Taylor series expansion of $f(y)$ and substituted in the original expansion yielding error of order $O(h^3)$. Use

$$f(y_0 + \Delta y) = f(y_0) + \Delta y f'(y_0) + O(\Delta y^2)$$

Choose $\Delta y = h f(y_0)$ to yield

$$f(y_0 + h f(y_0) ) = f(y_0) + h f(y_0) f'(y_0) + O(\Delta y^2)$$

Use $f(y_0) f'(y_0) = y^{''}_0$ and multiply by $\frac{h}{2}$ to obtain

$$\frac{h}{2} f( y_0 + h f(y_0) ) - \frac{h}{2} f(y_0) = \frac{h^2}{2} y^{''}_0 + O(h^3)$$

Substituting back in the first Taylor series yields

$$y(t_0+h) = y(t_0) + \frac{h}{2} ( f(y_0) + f(y_0 + h f(y_0) )$$

This is the Predictor-Corrector Improved Euler method. It is of the same order as the Mid-Point method.