Computational LeapFrog

Equations (4) are called quasi-Hamiltonian because, even though they employ generalized velocities, they describe the motion in the space of canonical variables. Accordingly, numerical trajectories computed with appropriate integrators will conserve the symplectic structure. Eor example, an implicit leapfrog integrator can be expressed as... 

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 (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... 

In general, boundary conditions are imposed at the end of each stage of (2.41) or the leapfrog time-step. Finally, in the case of absorbing boundary conditions, all derivatives are computed by the implicit algorithm across the entire domain including its interior and the absorber. Then, each system is updated by (2.41). 

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. 

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. 

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

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. 

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. 

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 ... 

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,... 

The solution of this set of equations is managed by a computer algorithm. The most common is the so-called leapfrog , which works stepwise by ... 

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