Euler method

Euler integration is the most basic kind of numerical integration for calculating trajectories from forces at discrete timesteps.

Derivation

Euler integration is simply derived from equations for the derivatives of the position and velocity of an object.

$a(t)={\frac {dv(t)}{dt}}$

and

$v(t)={\frac {dx(t)}{dt}}$

become

$v(t_{0}+\Delta t)=v(t_{0})+\Delta ta(t_{0})$

and

$x(t_{0}+\Delta t)=x(t_{0})+\Delta tv(t_{0})$

Error

The magnitude of the errors arising from Euler integration can best be demonstrated by comparison to a Taylor expansion of the trajectory of an object. If we assume that a(t), v(t) and x(t) are all known exactly at a time $t_{0}$, then Euler integration gives the position at time $t_{0}+\Delta t$ as:

$x(t_{0}+\Delta t)=x(t_{0})+\Delta tv(t_{0})+\Delta t^{2}a(t_{0})$

In comparison, the Taylor expansion of the trajectory gives:

$x(t_{0}+\Delta t)=x(t_{0})+\Delta tv(t_{0})+{\frac {1}{2}}\Delta t^{2}a(t_{0})+O(3)$

The error introduced by Euler integration is thus given by the difference between these equations:

${\frac {1}{2}}\Delta t^{2}a(t_{0})+O(3)$

Even if the $\Delta t^{2}$ term is removed through a common adjustment to the Euler integrator, the error still contains third-order terms in $\Delta t$. This make Euler integration less accurate than Verlet integration or Runge-Kutta integration, which have errors of fourth- and fifth-order respectively.

See Also

Molecular Dynamics Verlet integration