Geometric numerical integration illustrated by the Störmer–Verlet method

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Generalizing Hamiltonian Monte Carlo with Neural Networks (2017, methodology): “The dynamics are typically simulated using the leapfrog integrator (Hairer et al., 2003; Leimkuhler & Reich, 2004), which for a single time step consists of:”
A simple and effective Verlet-type algorithm for simulating Langevin dynamics (2012, methodology): “…two successive time steps and eliminating the velocity variables from the equations, the more original form of the St¨ormer-Verlet method [3, 4] (here called the position-Verlet method), is found with the associated velocity calculated by the central difference The Verlet scheme is…”
Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites (2012, methodology): “We employ the explicit time integration Velocity-Verlet algorithm which is a more numerically stable version of centraldifferences [15, 41].”
Hamiltonian Monte Carlo Without Detailed Balance (2014, background): “Leapfrog, or Störmer-Verlet, integration provides a discrete time approximation to Hamiltonian dynamics (Hairer et al., 2003).”
Studies of dynamic crack propagation and crack branching with peridynamics (2010, unknown): “The Velocity–Verlet algorithm (Hairer et al. 2003) is: Figure 4.”