Verlet algorithm

The Verlet algorithm is a method for the numerical solution of Newton's equations of motion . It arises from the Leapfrog method by eliminating the speed calculations. The Verlet algorithm is often used in molecular dynamics simulations in theoretical chemistry . The algorithm is named after Loup Verlet .

Derivation

Two third order Taylor expansions of the position are established . One of them is developed forwards and one backwards in time. Here is the speed and the acceleration. ${\displaystyle {\vec {x}}(t)}$ ${\displaystyle (t+\Delta t)}$ ${\displaystyle (t-\Delta t)}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {a}}}$

${\displaystyle {\vec {x}}(t+\Delta t)={\vec {x}}(t)+{\vec {v}}(t)\Delta t+{\frac {1}{2}}{\vec {a}}(t)\Delta t^{2}+{\frac {1}{6}}{\vec {b}}\Delta t^{3}+O(\Delta t^{4})}$

${\displaystyle {\vec {x}}(t-\Delta t)={\vec {x}}(t)-{\vec {v}}(t)\Delta t+{\frac {1}{2}}{\vec {a}}(t)\Delta t^{2}-{\frac {1}{6}}{\vec {b}}\Delta t^{3}+O(\Delta t^{4})}$

Adding the two equations gives

${\displaystyle {\vec {x}}(t+\Delta t)=2{\vec {x}}(t)-{\vec {x}}(t-\Delta t)+{\vec {a}}(t)\Delta t^{2}+O(\Delta t^{4})}$.

This is the general equation of the Verlet algorithm. The acceleration depends on the potential and the mass of the particle , it can with ${\displaystyle {\vec {a}}(t)}$ ${\displaystyle V}$ ${\displaystyle m}$

${\displaystyle {\vec {a}}(t)=-{\frac {1}{m}}\nabla V(x(t))={\frac {\vec {F}}{m}}}$

to be determined.

application

If the position is known, first, the position has more than ${\displaystyle {\vec {x}}_{0}}$ ${\displaystyle {\vec {x}}_{1}}$

${\displaystyle {\vec {x}}_{1}={\vec {x}}_{0}+{\vec {v}}_{0}\Delta t+{\frac {1}{2}}{\vec {a}}\Delta t^{2}}$

to be determined. If the positions and are then known, all of the following can be determined by iterating the Verlet algorithm . ${\displaystyle {\vec {x}}_{0}}$ ${\displaystyle {\vec {x}}_{1}}$ ${\displaystyle {\vec {x}}_{n}}$

${\displaystyle {\vec {x}}_{n+1}=2{\vec {x}}_{n}-{\vec {x}}_{n-1}+{\vec {a}}_{n}\Delta t^{2}}$

The Verlet algorithm only provides positions and no speeds. These must therefore be determined separately in order to calculate from them and thus the total energy. This is necessary to check the conservation of energy. ${\displaystyle E_{kin}}$

Web links

• Alexander Fulst, Christian Schwermann: Molecular dynamics simulation. Pp. 7-10.
• Thomas Jacobsen: Advanced Character Physics. In: Gamasutra. (English)