Leapfrog method

Update for positions:

x_i^m+1 = x_i^m + h v_i^m+1/2

v_i^m+1/2 = v_i^m-1/2 + h a_i^m

One immediate problem is how to begin: the first update of positions from x_i⁰ to x_i¹ requires v_i^1/2. One way around this is to use a simple Euler half-step to approximate the needed velocity:

v_i^1/2 = v_i⁰ + h/2 a_i⁰

Verlet published a paper in 1967 ("Computer Experiments on Classical Fluids" in Physical Review 159(1), 98--103, 1967) that described a method like this; some people call this the Verlet method. However, Stormer had published virtually the same algorithm in 1907 and had analyzed a family of algorithms based on a certain generalized condition; thus this method is sometimes known as one of the Stormer methods.

Quoting Greenspan: The name "leap frog" derives from the way position and velocity are defined at alternate, sequential time values ... symboliz[ing] the children's game "leap frog".

References

• Appendix III of "N-body problems and models" by Donald Greenspan, c2004
• Section I.1.4 of "Geometric Numerical Integration: Structure-preserving algorithms for ordinary differential equations" by Hairer, Lubich, and Wanner, 2006
• see also the second half of section 1.3.4 of "Numerical Solutions of the N-body Problem" by Andrzej Marciniak, 1985.
• Silly-pedia
• in Hairer, Norsett, and Wanner's "Solving Ordinary Differential Equations I: Nonstiff Problems": Section III.10
• not in the Burden and Faires "Numerical Analysis" textbook