Integration algorithms stability

It is tempting to get an improved value x 0) by extrapolation of x 2h) and x h) to h = 0. However, in general, if this improved value is introduced into the integration algorithm, the properties of stability disappear. Thus, this extrapolation process can only be used at the end of computations and is called global extrapolation (or passive extrapolation). 

To illustrate the difference in stability properties between explicit and implicit integration algorithms, consider again the equation used to describe valve dynamics in Section 2.2. Dropping the subscripts from equation (2.9) for clarity and generality, and setting the demanded valve travel, xj, to zero, indicating a... 

In many practical applications, the integration of well-conditioned systems carried out using traditional methods requires a very small integration step to ensure algorithm stability. Such a step is generally much smaller than required to ensure reasonable method accuracy. 

Although qualitative, this discussion shows that controlling only the local error is usually a good way to keep the integration step within the region of algorithm stability since is overestimated as it exits such a region. 

In MD simulations, the classical Newtonian equations of motion are numerically integrated for all particles in the simulation box. The size of the time step for integration depends on a number of factors, including temperature and density, masses of the particles and the nature of the interparticle potential, and the general numeric stability of the integration algorithm. In the MD simulations of aqueous systems, the time step is typically of the order of femtoseconds (10 s), and the dynamic trajectories of the... 

M. Klisinski, A. Mostrom, On stability of multi-time step integration algorithm, J. Eng. Mech. ASCE, 124 (1998) 783. 

In the sections that follow, we examine systematically the error propagation and stability of several numerical integration methods and suggest ways of reducing these errors by the appropriate choice of step size and integration algorithm. 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

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. 

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