Leap-frog

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

An example of a symplectic/time-reversible method is the Verlet (leap-frog) scheme. This method is applicable to separataP Hamiltonian systems of the... 

HyperChem employs the leap frog algorith m to integrate the et uaLioMs of motion. Th is algoritlim updates the positions of atom s and the velocities for the n e.x 1 time step by tli is ca leu la lion (equation 26). 

The leap-frog algorithm uses the simplest central difference I or-m n la for a derivative... 

As described previously, the Leap-frog algorithm for molecular dynamics requires an initial configuration for the atoms and an initial set of velocity vectors. /2- These initial velocities can come... 

Figure 1 A stepwise view of the Verlet integration algorithm and its variants, (a) The basic Verlet method, (b) Leap-frog integration, (c) Velocity Verlet integration. At each algorithm dark and light gray cells indicate the initial and calculating variables, respectively. The numbers in the cells represent the orders m the calculation procedures. The arrows point from the data that are used in the calculation of the variable that is being calculated at each step.

Modifications to the basic Verlet scheme have been proposed to tackle the above deficiencies, particularly to improve the velocity evaluation. One of these modifications is the leap-frog algorithm, so called for its half-step scheme Velocities are evaluated at the midpoint of the position evaluation and vice versa [12,13]. The algorithm can be written as... 

Figure lb gives a graphical representation of the steps involved in the leap-frog propagation. The current velocity v , which is necessary for calculating the kinetic energy, can be calculated as... 

The main disadvantage of this algorithm is that it is computationally a little more expensive than the simpler Verlet or leap-frog algorithms (though the added accuracy often outweighs this slight overhead). 

The numerical aspect, and the lack of explicit velocities, in the Verlet algorithm can be remedied by the leap-frog algorithm. Performing expansions analogous to eqs. (16.28) and (16.29) with half a time step followed by subtraction gives... 

Lean air/fuel mixtures, 10 36-37 Leap-frog technique, sampling via, 26 1016... 

Stellar nucleosynthesis thereby leap-frogs over three fragile elements, lithium, beryllium and boron, moving more or less directly from helium to... 

Development of road maps or structured innovation plans would probably work for parametric improvements around the structure, but the basic structure itself is unpredictable and cannot be regulated because the future cannot be predicted. Road maps can plan the incremental improvements, which lead to important changes, but they cannot plan breakthrough or leap frog types of discoveries. 

The above molecular dynamics equations are then solved using the the standard and robust leap-frog algorithm [29]. 

The interesting thing is that all but the last equation are leap-frog forms, which by themselves result in an unstable solution the mere addition of the last (4.39) renders the system stable, and the solution is of 0(8t2). 

---