Leapfrog method

The Leapfrog method is a simple method of numerically integrating an ordinary differential equation of the type

${\displaystyle {\ddot {x}}={\dot {v}}=a(x)}$.

or more generally of conservative systems that follow Newton's second axiom of classical dynamics and describe, for example, the movement of one or more objects in a potential field: ${\displaystyle {\ddot {\mathbf {x} }}=-{\frac {\mathrm {grad} \,V(\mathbf {x} )}{M}}}$ ${\displaystyle V}$

${\displaystyle {\ddot {\vec {x}}}_{k}=-{\frac {1}{m_{k}}}\nabla _{{\vec {x}}_{k}}V\left({\vec {x}}_{1},\dots ,{\vec {x}}_{N}\right),\quad k=1,\dots ,N.}$

Leapfrog integration is a second order method and therefore generally delivers more accurate results than, for example, Euler's polygon method , which is only of first order. In addition, it is invariant under time reversal and, in physical problems, receives quantities such as momentum and angular momentum , which are also conserved quantities of the original system, exactly. Furthermore, a disturbed energy function in order 3 is obtained, while the method has the global convergence order 2 .

Representation as a leapfrog method

The leapfrog integration calculated alternately, the positions and the velocities at different times, similar to the Leapfrog (engl. Leapfrog). The step equations for the procedure are: ${\displaystyle x}$ ${\displaystyle v}$

${\displaystyle {\begin{aligned}x_{i+1}&=x_{i}+v_{i+1/2}\,\Delta t\\[.3em]v_{i+3/2}&=v_{i+1/2}+a(x_{i+1})\,\Delta t\end{aligned}}}$

with the start values and . ${\displaystyle x_{0}}$ ${\displaystyle v_{1/2}}$

Representation as a one-step process

By linear interpolation of intermediate values , the Leapfrog method can be viewed as a combination of the two variants of the symplectic Euler method :

${\displaystyle {\begin{aligned}{\text{(SE1)}}\quad &\left\{{\begin{array}{rcl}v_{i+1/2}&=&v_{i}+a(x_{i})\,\Delta t/2\\[.5em]x_{i+1/2}&=&x_{i}+v_{i+1/2}\,\Delta t/2\end{array}}\right.\\[.7em]{\text{(SE2)}}\quad &\left\{{\begin{array}{rcl}x_{i+1}&=&x_{i+1/2}+v_{i+1/2}\,\Delta t/2\\[.5em]v_{i+1}&=&v_{i+1/2}+a(x_{i+1})\,\Delta t/2\end{array}}\right.\\\end{aligned}}}$

Every single step and thus the composition is a symplectic transformation and therefore receives volumes in the phase space . This also results in the exact conservation of momentum and angular velocity, insofar as the exact system receives them.

Inserting (SE1) and (SE2) into one another leads to:

${\displaystyle {\begin{array}{rcl}x_{i+1}&=&x_{i}+v_{i}\Delta t+a_{i}(x_{i}){\frac {\Delta t^{2}}{2}}\\v_{i+1}&=&v_{i}+\Delta t{\frac {a(x_{i+1})+a(x_{i})}{2}}\end{array}}}$

That means that the new location is projected by Taylor expansion up to the second order starting from the old location in the phase space and the new speed up to the first order; however, with a modified acceleration which is the mean value of the two accelerations at times and . ${\displaystyle t_{i}}$ ${\displaystyle t_{i+1}}$

Representation as a multi-step process

If you eliminate the speed calculations from the Leapfrog version, the result is

${\displaystyle {\begin{aligned}(x_{i+1}-x_{i})-(x_{i}-x_{i-1})&=(v_{i+1/2}-v_{i-1/2})\,\Delta t\\[.3em]x_{i+1}-2x_{i}+x_{i-1}&=a(x_{i})\,\Delta t^{2}\\[.7em]\implies \quad x_{i+1}=2x_{i}-x_{i-1}&+a(x_{i})\,\Delta t^{2},{\text{ mit Startwerten }}x_{0},x_{1}=x_{0}+v_{0}\,\Delta t+{\tfrac {1}{2}}a(x_{0})\,\Delta t^{2}\end{aligned}}}$

the Verlet method, which can also be derived directly from as a symmetrical discretization . This discretization has a local error of and thus (because of the double integration) a global error of the size for the difference between the exact and approximate solution at the end time . Another variant is known under the name Velocity-Verlet. ${\displaystyle {\ddot {x}}=a(x)}$ ${\displaystyle O(\Delta t^{4})}$ ${\displaystyle O(t_{N}^{2}\Delta t^{2})}$ ${\displaystyle x(t_{N})-x_{N}}$ ${\displaystyle t_{N}=t_{0}+N\Delta t}$

history

A first description of this process was found by Richard Feynman in Isaac Newton's Principia of 1687 in an argument to derive Kepler's laws from the equations of motion. Variants of this process were used by JF Encke in 1860 and by C. Störmer in 1907, among others.

example

If you consider the oscillation equation with the exact solution , you get the transition for in the one-step formulation ${\displaystyle {\ddot {x}}+x=0}$ ${\displaystyle x(t)=C\,\cos(t+\phi )}$ ${\displaystyle {\dot {x}}=v,\;{\dot {v}}=-x}$

${\displaystyle {\binom {x_{k+1}}{v_{k+1}}}={\begin{pmatrix}1-{\tfrac {1}{2}}\Delta t^{2}&\Delta t\\-\Delta t(1-{\tfrac {1}{4}}\Delta t^{2})&1-{\tfrac {1}{2}}\Delta t^{2}\end{pmatrix}}{\binom {x_{k}}{v_{k}}}.}$

It turns out that the modified energy functional is exactly obtained. More precisely , the error in the energy has a global limit of order . The approximate solution runs for all times on the ellipses defined by the constant modified energy level. ${\displaystyle E_{\Delta t}(v,x)={\tfrac {1}{2}}\left(v^{2}+(1-{\tfrac {1}{4}}\Delta t^{2})x^{2}\right)}$ ${\displaystyle E_{\Delta t}(v_{n},x_{n})=E(v_{0},x_{0}).}$ ${\displaystyle O(\Delta t^{2})}$

swell

1. ↑ Michael Griebel, Stefan Knapek, Gerhard Zumbusch, Attila Caglar: Numerical Simulation in Molecular Dynamics . Springer, 2004.
2. ^ Ernst Hairer, Christian Lubich, Gerhard Wanner: Geometric numerical integration illustrated by the Störmer / Verlet method . In: Acta Numerica . 12, 2003, pp. 399-450. doi : 10.1017 / S0962492902000144 .