Semi-implicit Euler method

The Euler-Cromer algorithm is a modification of the Euler approximation for numerical integration which gives much better results for oscillatory solutions.

Given a pair of differential equations of the form

${dx \over dt}=v(x,t)$

${dv \over dt}=a(x,v,t)$

and initial conditions

$(x_{0},v_{0})\quad$

an approximate discrete solution is given by

$v_{n+1}=v_{n}+a_{n}\Delta t\quad$

$x_{n+1}=x_{n}+v_{n+1}\Delta t\quad$

where $\Delta t$ is the timestep.