Leapfrog method

The calculation of A x) and Ax x ) can be done in a systematic manner. First the calculation of A x) is coded, and then this is differentiated with respect to each of the components of x to yield code for Ax x). An example of this procedure for the leapfrog method is given in Appendix B. 

As an example suppose that the leapfrog method with time step St is coded for the calculation of A x). This is then differentiated to obtain Ax x). The result is the following code for calculating A x) and Ax x) Initialization is given by... 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

The method is based on another time-marching scheme not mentioned in the above sections the leapfrog method, using central differences. Equation (4.1) can be approximated as... 

Kimble and White were aware that leapfrog methods are unstable and simply remark that this did not seem to apply to their method. Also, they mention the use of 5 points for all approximations but their table of discretisations shows that they used 6 points at the edges for the spatial second derivative. This is no doubt because, as Collatz already mentions in 1960 [170], the asymmetric 5-point second derivative is only third-order, while a 6-point formula is fourth-order, like the symmetrical 5-point ones used in the bulk of the grid. So, for the second spatial derivative at index i = 1, the form 2/2(6) was used, and the reverse, form 2/5 (6) at i = N. 

For example, the second order (two-step) Leapfrog method frequently used in meteorology and oceanography can be deduced from the midpoint rule [66, 49, 158] ... 

Cycle Division. In one popular scheme the same iteration process is used on the different processes, but with widely separated seeds on each process. There are two related schemes (a) the leapfrog method, where processor i gets (, ui+M,. .., where M is the total number of processes (e.g., process 1 gets the first member of the sequence, process 2 the second, and so forth) and (b) in the cycle splitting method, where process i + 1 gets uit/Af, + . .., where L is the cycle length and M is the number of processes. (That is, the first process will get the first L/M numbers, the second process the second L/M numbers, and so forth.)... 

Statistical tests performed on the original sequence are not necessarily adequate for the divided sequence. For example, in the leapfrog method correlations M numbers apart in the original sequence become adjacent correlations of the split sequence. 

Coupling this to the stochastic evolution computed from solving the OU parts in both and p yields an alternative to Langevin dynamics that is 2nd order accurate in its steady state distribution and long term averages. Let us refer to this as the stochastic line sampling with leapfrog method. 

We used the results of a molecular dynamics simulation to interpret the II-A isotherms. The evolution of the many-particle system can be described by integrating Newton s equations of motion. To integrate the differential equation system we used the leapfrog method [40]. The simulation of the compression was performed for 1,000 particles in a rectangular cell with periodic boundary conditions. The size distribution of the particles could be set... 

The Verlet velocity algorithm overcomes the out-of-synchrony shortcoming of the Verlet leapfrog method. The advantage here is that the positions, velocities, and accelerations are computed at the same time t. There is no compromise on precision. The Verlet velocity algorithm is as follows ... 

Two methods for measuring potential are commonly used a single electrode potential and a double electrode potential measurement. These methods have been described in detail in Chapter 5. The principles are the same as applied to the measurement of potential in reinforcing steels. The double electrode method, also called leapfrog method, is preferable in measurement... 

Few-body problems can be handled by conventional integrators, such as Runge-Kutta or Adams-Moulton methods. Here one calculates the position and velocity for each particle and then the precise two-body interaction for that body with every other particle in the system. Both methods are predictor-corrector procedures in which the next step is computed and corrected iteratively. Leapfrog methods, which use the velocity from one step and the positions from the previous step to compute the new positions, are also computationally efficient and stable. The basic problem is to solve the equations of motion for a particle at position Fj,... 

There are various, essentially equivalent, versions of the Verlet algoritlnn, including the origmal method employed by Verlet [13, 44] in his investigations of die properties of the Lennard-Jones fluid, and a leapfrog fonn [45]. Here we concentrate on the velocity Verlet algoritlnn [46], which may be written... 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

The (ode-) method called leapfrog has been mentioned in Chap. 4, where (4.38) describes it. This was used by Richardson [468] to solve a parabolic pde, apparently with success. The computational molecule corresponding to this method is... 

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