Numerical analysis Ordinary differential equations Proposition

Beeman's algorithm

Builds on Initial value problem Verlet integration

Statement

Let $a \colon \R^d \to \R^d$ be continuous, and consider the equation $x'' = a(x)$ . Fix a step size $h > 0$ and write $a_n = a(x_n)$ . Beeman's algorithm computes, from $x_n$ , $v_n$ , $a_n$ and $a_{n-1}$ , $x_{n+1} = x_n + h v_n + \frac{h^2}{6} (4 a_n - a_{n-1}),$ $v_{n+1} = v_n + \frac{h}{6} (2 a_{n+1} + 5 a_n - a_{n-1}).$ Then for all $n \ge 2$ for which the sequences are defined, $x_{n+1} = 2 x_n - x_{n-1} + h^2 a_n.$ So Beeman's algorithm produces the same positions as the Störmer–Verlet method, once both start from the same $x_1$ and $x_2$ .

Remarks

Subtract the position formula at step $n - 1$ from the one at step $n$ : $x_{n+1} - 2x_n + x_{n-1} = h (v_n - v_{n-1}) + \frac{h^2}{6} (4 a_n - 5 a_{n-1} + a_{n-2}).$ The velocity formula gives $h (v_n - v_{n-1}) = \frac{h^2}{6}(2 a_n + 5 a_{n-1} - a_{n-2})$ , and the sum is $h^2 a_n$ . The difference from Verlet lies only in the velocities, which Beeman's formula computes more accurately than the central difference $(x_{n+1} - x_{n-1})/(2h)$ ; this improves the computed kinetic energy. The velocity formula above is the implicit form; a common explicit variant predicts $v_{n+1}$ first when the force depends on the velocity.

Going deeper

For a student

The identity shows that Beeman's method inherits the stability and the long-time behaviour of Verlet's positions. It is a multistep method, so it needs a starting procedure to produce $a_{-1}$ or $x_1$ . In practice velocity Verlet is more common, because it gives similar accuracy with less storage.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`RHQ88QD2BE`
URI	`m2://RHQ88QD2BE`
Kind	statement, proposition
Writing	W0 stub
MSC	65L06, 70-08
Status	This copy of the site has no build data. Run `m2 status RHQ88QD2BE`.
About	Verlet integration
Uses	Initial value problem, Verlet integration