Leapfrog integration

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, leapfrog integration is a <a href="/facts/Numerical_methods_for_ordinary_differential_equations/rbrYBco3">method</a> for numerically integrating <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> of the form

x
              ¨
            
          
        
        =
        
          
            
              
                d
                
                  2
                
              
              x
            
            
              d
              
                t
                
                  2
                
              
            
          
        
        =
        A
        (
        x
        )
        ,
      
    
    {\displaystyle {\ddot {x}}={\frac {d^{2}x}{dt^{2}}}=A(x),}

or equivalently of the form

v
              ˙
            
          
        
        =
        
          
            
              d
              v
            
            
              d
              t
            
          
        
        =
        A
        (
        x
        )
        ,
        
        
          
            
              x
              ˙
            
          
        
        =
        
          
            
              d
              x
            
            
              d
              t
            
          
        
        =
        v
        ,
      
    
    {\displaystyle {\dot {v}}={\frac {dv}{dt}}=A(x),\qquad {\dot {x}}={\frac {dx}{dt}}=v,}

particularly in the case of a <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>.

The method is known by different names in different disciplines. In particular, it is similar to the velocity Verlet method, which is a variant of <a href="/facts/Verlet_integration/Wtzt8hkX">Verlet integration</a>. Leapfrog integration is equivalent to updating positions 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 and velocities 
 
 
 
 v
 (
 t
 )
 =
 
 
 
 x
 ˙
 
 
 
 (
 t
 )
 
 
 {\displaystyle v(t)={\dot {x}}(t)}
 
 at different interleaved time points, staggered in such a way that they "<a href="/facts/Leapfrog/XKN9wxfy">leapfrog</a>" over each other.
Leapfrog integration is a <a href="/facts/Differential_equation/9MGbE3Ca">second-order</a> method, in contrast to <a href="/facts/Euler_integration/7yF5B4z9">Euler integration</a>, which is only first-order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step 
 
 
 
 Δ
 t
 
 
 {\displaystyle \Delta t}
 
 is constant, and 
 
 
 
 Δ
 t
 <
 2
 
 /
 
 ω
 
 
 {\displaystyle \Delta t<2/\omega }
 
.
Using Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated.

Leapfrog integration open-in-new

Leapfrog integration