Euler-Cromer Method

Simple modification of Eqs. (5), (4) yields solutions that are stable for oscillatory systems. Such Euler-Cromer algorithm or last-point approximation leads to the following difference equations

$\displaystyle v_{n+1}$ $\textstyle =$ $\displaystyle v_n + a_n \Delta t,$ (6)
$\displaystyle x_{n+1}$ $\textstyle =$ $\displaystyle x_n + v_{n+1} \Delta t.$ (7)

As pointed out by Cromer in: A. Cromer, Stable solutions using the Euler Approximation, American Journal of Physics, 49, 455 (1981), this simple modification conserves energy for oscillatory problems (unlike Euler method which artifactually increases energy of the oscillator with time).