Leapfrog integration
The leapfrog scheme is a simple method for the numerical integration of an ordinary differential equation of the type
Or more generally of conservative systems of the classical dynamics, for example, describe the movement of one or more objects in a potential field V,
The Leapfrog integration is a second-order method, and therefore generally provides more accurate results than for example the Euler Polygonzugverfahren, which is only of first order. Moreover, it is invariant under time reversal and gets into physical problems sizes as the momentum and angular momentum, which are also conserved quantities of the original system, exactly. Furthermore, get a disturbed energy function in order 3, while the method has the global convergence order 2.
View as leapfrog scheme
The Leapfrog integration alternately calculated the positions and velocities at different times, similar to the Leapfrog (English leapfrog ). The step equations for the method are:
With the starting values and.
View as step method
By linear interpolation of intermediate values, the Leapfrog method can be seen as a combination of two variations of the Euler method symplectic:
Each step, and thus the composition is a symplectic transformation and therefore receives volumes in phase space. It also follows the exact conservation of momentum and angular velocity, as far as the exact system receives this.
View as multistep methods
After eliminating from the Leapfrog version of the velocity calculations, we obtain
The Verlet method can be derived as symmetric discretization directly from also. This discretization has a local error of and therefore (because of the double integration ) a global error of the size of the difference between exact and approximate solution to stop time. Another variation is known as the Velocity Verlet.
History
A first description of this method was found by Richard Feynman in Isaac Newton's Principia of 1687 in an argument for the derivation of Kepler's laws from the equations of motion. Among other variants of this process in 1860 by JF Encke and 1907 by C. Störmer were used.
Example
If we consider the wave equation with the exact solution is obtained for the one-step formulation of the transition
It turns out that the modified energy functional is obtained exactly. More precisely, thus the error in the energy has a global barrier of order. The approximate solution runs for all times on the ellipses defined by the modified constant energy level.
Swell
- Ordinary Differential Equations