4.2.1 Verlet Method

Apply DDST,

$$\left. \frac{d^{2}y}{dt^{2}} \right\vert _{n}=\frac{\delta^{2}y_{n}}{(\Delta t)^{2}} +O[(\Delta t)^{2}]$$

to find

$$\fbox{$ y_{n+1}=2y_{n}-y_{n-1}+b_{n}(\Delta t)^{2}+O[(\Delta t)^{4}] $}$$

Note: velocity $v \equiv \dot{y}$ does not appear explicitly. Use the crude estimate

$$v_{n} = \frac{1}{2\Delta t}[y_{n+1}-y_{n-1}]+O[(\Delta t)^{2}]$$

Stability of the Verlet scheme:

Harmonic oscillator:

$$y_{n+1}=2y_{n}-y_{n-1}-\omega_{0}^{2}y_{n}(\Delta t)^{2}$$

leads to

$$g^{2}-(2-\alpha^{2})g+1=0$$

with $\alpha\equiv \omega_{0} \Delta t$. The root

$$g=(1-\frac{\alpha^{2}}{2})\pm \sqrt{\frac{\alpha^{4}}{4}-\alpha^{2}}$$

is imaginary for $\alpha <2$, with $\vert g\vert^{2}=1$.

Equivalent formulations: Verlet leapfrog and Velocity Verlet.





Leapfrog version of the Verlet method



Swope's formulation of the Verlet algorithm