Math 2.0

Numerical analysis Ordinary differential equations Proposition

Leapfrog integration

Builds on Hamiltonian systemSymplectic integratorInitial value problem

Statement

Let V ⁣:ℝd→ℝV \colon \R^d \to \R be three times continuously differentiable, put a=−∇Va = -\nabla V, and consider the equation x′′=a(x)x'' = a(x) with x(0)=x0x(0) = x_0 and x′(0)=v0x'(0) = v_0. Fix a step size h>0h > 0. The leapfrog method, in its kick-drift-kick form, maps (xn,vn)(x_n, v_n) to (xn+1,vn+1)(x_{n+1}, v_{n+1}) by vn+1/2=vn+h2a(xn),xn+1=xn+h vn+1/2,vn+1=vn+1/2+h2a(xn+1).v_{n+1/2} = v_n + \tfrac{h}{2} a(x_n), \qquad x_{n+1} = x_n + h\, v_{n+1/2}, \qquad v_{n+1} = v_{n+1/2} + \tfrac{h}{2} a(x_{n+1}). Let Φh ⁣:ℝ2d→ℝ2d\Phi_h \colon \R^{2d} \to \R^{2d} denote this map.

  1. Φh\Phi_h is symplectic: it preserves the symplectic form ∑i=1ddxi∧dvi\sum_{i=1}^d dx_i \wedge dv_i.
  2. Φh\Phi_h is time-reversible: Φ−h(Φh(x,v))=(x,v)\Phi_{-h}(\Phi_h(x, v)) = (x, v) for all (x,v)(x, v).
  3. The method has order 22: on a bounded time interval [0,T][0, T], the error max⁡nh≤T∣(xn,vn)−(x(nh),x′(nh))∣\max_{nh \le T} \abs{(x_n, v_n) - (x(nh), x'(nh))} is O(h2)O(h^2) as h→0h \to 0.

Remarks

The equation is the Hamiltonian system with H(x,v)=12∣v∣2+V(x)H(x, v) = \frac12 \abs{v}^2 + V(x). Each of the three substeps is the exact flow, for time h/2h/2, hh or h/2h/2, of the Hamiltonian V(x)V(x) or 12∣v∣2\frac12\abs{v}^2. Exact Hamiltonian flows are symplectic, and so is their composition, which gives clause 1. The composition is symmetric, which gives clause 2, and a symmetric method has even order, which together with consistency gives clause 3. So leapfrog is a symplectic integrator, and its energy error stays bounded over very long times instead of growing. Example: for a(x)=−xa(x) = -x in one dimension, Φh\Phi_h is a linear map with determinant 11 and trace 2−h22 - h^2. For 0<h<20 < h < 2 its eigenvalues are distinct and lie on the unit circle, so all orbits stay bounded.

Going deeper

For a student

Leapfrog is the Störmer–Verlet method written in one-step form, and it is the standard integrator of molecular dynamics and of NN-body simulations. It is the simplest example of a splitting method: it is the Strang composition of the kinetic and potential flows. Higher-order symplectic methods are often built by composing leapfrog steps with well-chosen step sizes.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Leapfrog_integration
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