In mathematics and computational science, Euler integration is the most basic kind of numerical integration for calculating trajectories from forces at discrete timesteps. More generally, the method is a numerical procedure for solving first-order differential equations with a given initial value.

Derivation

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

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

and

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

become

$v(t\_0\; +\; \backslash Delta\; t)\; =\; v(t\_0)\; +\; \backslash Delta\; t\; a(t\_0)\backslash ,$

and

$x(t\_0\; +\; \backslash Delta\; t)\; =\; x(t\_0)\; +\; \backslash Delta\; t\; v(t\_0)\backslash ,$

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 $t\_0\; +\; \backslash Delta\; t$ as:

$x(t\_0\; +\; \backslash Delta\; t)\; =\; x(t\_0)\; +\; \backslash Delta\; t\; v(t\_0)\; +\; \backslash Delta\; t^2\; a(t\_0)\backslash ,$

In comparison, the Taylor expansion of the trajectory gives:

$x(t\_0\; +\; \backslash Delta\; t)\; =\; x(t\_0)\; +\; \backslash Delta\; t\; v(t\_0)\; +\; \backslash frac\{1\}\{2\}\backslash Delta\; t^2\; a(t\_0)\; +\; O(\backslash Delta\; t^3)$

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

$-\backslash frac\{1\}\{2\}\backslash Delta\; t^2\; a(t\_0)\; +\; O(\backslash Delta\; t^3)$

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

There are some drawbacks when considering use of this method. The error from Euler's method is quite large. In addition to this, the method can be numerically unstable, especially for stiff equations. On the positive side, Euler's method is very simple to implement.

See also

• Molecular dynamics
• Verlet integration
• Leonhard Euler

Numerical analysis

Eulersches Polygonzugverfahren | Méthode d'Euler | 오일러 방법