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Euler's method

Euler's explicit method solves an ODE $y' = f(t, y)$ numerically by following the slope from each point a small step $h$ forward. It is simple but low order: the global error is $O(h)$ , so short steps are needed for good accuracy.

y_{n+1} = y_n + h\,f(t_n, y_n)

one Euler step

Symbols

$h$	step size
$f(t_n, y_n)$	the slope at $(t_n, y_n)$

Example

$y' = y$ , $y_0 = 2$ , $h = 0.1$ : each step is multiplied by $1.1$ .

$y_2 = 2\cdot 1.1^2 = 2.42$ .

Halving

h

roughly halves the global error — Euler only has first-order accuracy.

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Part of Numerical Methods: Differential equations and stability.