Leapfrog scheme

H. Spachmann, R. Schuhmann, and T. Weiland, Convergence, stability and dispersion analysis of higher order leapfrog schemes for Maxwell s equations, in Proc. 17th Ann. Rev. Prog. Appl. Comput. Electromagn., Monterey, CA, Mar. 2001, pp. 655-662. 

Vectors 521, for ( = 1,2,3, in (3.56), are temporary storage quantities for the intermediate values of the progressive integration. In an analogous manner, the remaining part of the leapfrog scheme is completed as... 

This paragraph complements the analysis of Section 2.5 by presenting a technique for the construction of improved fourth-order spatial operators through the use of the discrete dispersion relation. Principally, the algorithm considers the ordinary leapfrog scheme for time marching, while it involves the parametric expression of (2.107) for spatial differentiation. By substituting plane-wave constituents in Maxwell s equations, the 2-D dispersion relation for an isotropic medium is... 

In other words, the only difference between (5.60) and the second-order leapfrog scheme is of the supplementary terms - depending on the value of k - which increase the computational overhead with the consecutive use of spatial operators at additional nodes. It is stressed that the presence of lossy materials receives an analogous treatment with the exception of more complicated expressions beyond the fourth order. 

Concerning the explicit update of Fa(u t) in (6.36), the following modified leapfrog scheme is utilized... 

Table 11 shows some typical time difference sehemes, used to solve Eq. (23) and corresponding amplification factors C applied to Eq. (25). We also indicate whether the scheme is explicit or implicit (see Section 111. A), whether it is first- or second-order in accirracy, and whether it is computationally stable. The Crank-Nicholson seheme is accurate and stable, but because of its implieit nature, its application can be time consirming. The leapfrog scheme is most common and stable provided that ft> Ar < 1, but because three time levels are involved, it prodnees two independent solutions. In the limit At 0, C" c"", and C" (-1)" ", so that the seeond solntion is a computational mode, while the first is the physieal mode. The computational mode is a false solntion and mnst be controlled to prevent it from overwhelming the physieal mode. 

To obtain bounded solutions of Eq. (31), we must require that the eigenvalues of G not exceed unity in absolute value. This is essentially the von Neumann stability condition for a finite difference scheme. To satisfy the computational stability condition for the leapfrog scheme, we see from Eq. (34) that... 

There is a large body of hterature describing a variety of difference schemes to solve equations of the same type as Eq. (30). Arty difference scheme for solving atmospheric equations has some deficiencies as well as merits. For example, the leapfrog scheme is perhaps the most important among exphcit schemes, but there is a problem as pointed out in Section V.A. Because the computational mesh can be divided into odd and even points and the difference calculations at the odd points proceed independently of those at the even points, careless application of this scheme can cause out-of-phase values at two consecutive grid points. 

We specify a time step, during which the forces are assumed to remain constant. Then r, and v, are updated. There are several schemes for this to overcome problems associated with finite rather than infinitesimal time steps. The force (and thus the acceleration) is assumed to remain constant throughout the time step At. For example, in the Leapfrog Verlet algorithm (e.g. Allen and Tildesley, Further reading),... 

Now we are able to substitute B/t) in Eq. 8 from Eq. 9. After replacing the acceleration Rj (t) with the force F/ (t) we finally obtain Eq. 6. There are several others algorithms to integrate the equations of motion (e.g., leapfrog, Verlet). The consequences of different equation of motion integration schemes with regard to AMD are discussed in the excellent review of Remler and Madden (54). 

The Verlet scheme propagates the position vector with no reference to the particle velocities. Thus, it is particularly advantageous when the position coordinates of phase space are of more interest than the momentum coordinates, e.g., when one is interested in some property that is independent of momentum. However, often one wants to control the simulation temperature. This can be accomplished by scaling the particle velocities so that the temperature, as defined by Eq. (3.18), remains constant (or changes in some defined manner), as described in more detail in Section 3.6.3. To propagate the position and velocity vectors in a coupled fashion, a modification of Verlet s approach called the leapfrog algorithm has been proposed. In this case, Taylor expansions of the position vector truncated at second order... 

In this study the numerical simulations were performed with a 3-D mechanistic global Cologne Model of the Middle Atmosphere (COMMA) based on the primitive equations expressed in spherical coordinates for the horizontal and log-pressure coordinates in the vertical direction. The model equations are solved on the basis of an explicit numerical scheme (leapfrog) with a fixed time step of 450 sec. To avoid separate evolution at even and odd time steps, a Robert time filter is used. 

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... 

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.)... 

In practice, we use a more accurate computational scheme called the leapfrog algorithm. 

The Clar structure thus has six extra bonding orbitals as compared with the Fries structure. When both bonding schemes are correlated, as illustrated in Fig. 6.11, this sextet must correlate with the anti-bonding half of the Fries stmcture. It will thus be placed on top of the Clar band, and actually be nearly non-bonding, forming six low-lying virtual orbitals, which explains the electron deficiency of the leapfrog fullerenes. Moreover, as the derivation shows, they transform exactly as rotations and translations. 

A typical explicit second-order difference scheme, called leapfrog, is... 

Two modifications of the Verlet scheme are of wide use. The first is the leapfrog algorithm [3] where positions and velocities are not ealeulated at the same time velocities are evaluated at half-integer time steps ... 

Figure 1. Schematic representation of a molecular dynamics simulation. The scheme for integrating the equations of motion is known as the leapfrog algorithm. The figure shows a flow diagram involving samples of the coordinates for a simulation of N steps.

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