Divergence oscillating

The explicit method is easy to use, but it suffers from an undesirable feature that Severely restricts its utility the explicit method is not unconditionally stable, and the largest permissible value of the lime step At is limited by the stability criterion. If the time step At is not sufficiently small, the solutions obtained by the explicit method may oscillate wildly and diverge from the actual solution. To avoid such divergent oscillations in nodal temperatures, the value of Af must be maintained below a certain upper limit established by the stability criterion. It can be shown mathematically or by a physical argument ba.sed pfl thc second law of thermodynamics tliat the stability criterion is satisfied if the coefficients of alt in the Tjj, expressions fcalled the primary... 

The reactor core is designed so its nuclear characteristics do not contribute to a divergent power transient. The reactor is designed so there is no tendency for divergent oscillation of any operating characteristics considering the interaction of the reactor with other appropriate plant systems. 

In the general sense, the term Instability is used to denote a state in a system wherein disturbances tend to grow In amplitude either exponentially or as a divergent oscillation. In determining the implications of instability of a design of an SGHWR system, there are three main modes that must be considered. 

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. 

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

Since the limit sum does not exist, the series is divergent. This is defined as a bounded or oscillating divergent series. Similarly for the value r = +1,... 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

Here (Oj is the excitation energy ErE0 and the sum runs over all excited states I of the system. From equation (5-37) we immediately see that the dynamic mean polarizability a(co) diverges for tOj=co, i. e has poles at the electronic excitation energies 0)j. The residues fj are the corresponding oscillator strengths. Translated into the Kohn-Sham scheme, the exact linear response can be expressed as the linear density response of a non-interacting... 

It is seen that the "electrochemical estimates of values of AG diverge from the straight line predicted from the harmonic oscillator model to a similar, albeit slightly smaller, extent than the experimental values. Admittedly, there is no particular justification for assuming that the reduction half reactions obey the harmonic oscillator model. However, it turns out that the estimates of AG are relatively insensitive to... 

The key problem is to find a method for making the new guess that converges rapidly to the correct answer. There are a host of techniques. Unfortunately there is no best method for all equations. Some methods that converge very rapidly for some equations will diverge for other equations i.e., the series of new guesses will oscillate around the correct solution with ever-increasing deviations. This is one kind of numerieal instability. 

The new guess can be simply the calculated value (this is called successive substitution). Convergence may be very slow because of (1) a very slow rate of approach of to or (2) an oscillation of back and forth around The loop can even diverge. 

Let us first consider the population probability of the initially excited adiabatic state of Model 1 depicted in Fig. 17. Within the first 20 fs, the quantum-mechanical result is seen to decay almost completely to zero. The result of the QCL calculation matches the quantum data only for about 10 fs and is then found to oscillate around the quantum result. A closer analysis of the calculation shows that this flaw of the QCL method is mainly caused by large momentum shifts associated with the divergence of the nonadiabatic couplings F = We therefore chose to resort to a simpler approximation... 

Here is seen the most obvious contrast with the kinematical theory. Both curves are plotted as a function of thickness in Figure 4.23. It is seen that the formulae give similar results for small thicknesses, that the kinematic theory diverges drastically at larger thicknesses, and that the dynamic intensity shows oscillations about a saturated level after a thickness of about. ... 

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