The Stability Threshold

Let us recall the method of studying the asymptotic numerical stability of a linear system 

The iteration is stable if m remains bounded as n condition for asymptotic stability  

This defines the stability region for Euler s method as a disk in the complex hX-plane centered around —1 of radius 1. This region is shown in Fig. 4.1. If hX, where A is an eigenvalue of A lies in the indicated stability region, then Euler s method will be stable for the linear system z = Az. 

If we consider applying this technique to the harmonic oscillator q = p p = —Q q, then the matrix A is 

Leimkuhler, C. Matthews, Molecular Dynamics, Interdisciplinary Applied 

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Figure 5.1.7a shows a side view of a lean propane flame, 10 cm in diameter, propagating downward in a top-hat flow. The flame speed is 9cm/s, below the stability threshold, and the flame is stable at all wavelengths. Figure 5.1.7b shows a near stoichiometric flame in the same burner. The flame is seen at an angle from underneath. The mixture is diluted with nitrogen gas to reduce to flame speed to the instability threshold (10.1 cm/s), so that the cells are linear in nature. The cell size here is 1.9 cm. Figure 5.1.7c shows a flame far above the instability threshold, the cell shape becomes cusped, and the cells move chaotically. 

If the driving force (Ay) is less than zero an oil-in-water dispersion forms, while if Ay is of the opposite sign an inverse (water-in-oil) dispersion is produced. The stability threshold represents a critical emulsifier concentration below which a kinetically stable macroemulsion is produced. These can be transformed,... 

Table 1 compares the Suspension, Emulsion and Microemulsion regimes. The level of surfactant with respect to the CMC and the stability threshold, the locus of nucleation, and the particle size clearly distinguish these three regimes. This table also illustrates the analogies between direct oil-in-water and inverse water-in-oil polymerization processes with respect to the aforementioned attributes. To summarize As the concentration of emulsifier increases, at a fixed temperature, monomer level, and aqueous and organic phase ratio, a transition between the suspension, emulsion and microemulsion domains ensues. This occurs with a corresponding decrease in particle size. Within the... 

This is the so-called linear stability condition of the Symplectic Euler method if hQ < 2 the integrator is stable. When hQ > 2, the eigenvalues of the discretization method are both real, with one strictly inside and one strictly outside the unit circle. This implies that the method will exhibit exponentially growing solutions. We say that the stability threshold of the Symplectic Euler method is 2/f2. 

The only way to increase the stability threshold is by introducing implicitness or by changing in some fundamental way the timestepping framework. We consider some alternatives below. 

The equations of DPD are a little more complicated than standard MD. It is therefore important to design accurate and efficient methods tailored to their special form. Most numerical methods for DPD simulation exhibit substantial statistical bias. Some methods have been developed to reduce the per timestep efficiency but at a price in terms of the conservation properties. The only way that the errors are tamed in some ad hoc schemes is by reducing the timestep size excessively (well below the stability threshold), but this may destroy the practical value of those methods. 

The techniques of direct electrochemistry are put to their best use in the study and manipulation of proteins for which the redox chemistry is not addressed effectively by other methods. Subjects to benefit particularly are proteins containing metal centres that may be intrinsically unstable or have redox chemistry at potentials beyond the stability threshold of the solvent system. Many proteins containing Fe-S clusters fall into this category. These centres are widely distributed in biological systems [164] where their most widely accepted role is as electron-transfer agents. 

The purpose of this paper is to indicate the effects of the variation of the cavity pressure and the movement of the cavity boundary on the dynamic oil film coefficients and the stability threshold in a plain journal bearing with an axial oil supply groove. 

Fig. 6 shows the stability thresholds for a horizontal rigid shaft. CP predicts higher critical speed than FC at low and high eccentricity ratio. 

Fig. 10 shows the stability thresholds for CP,CV and AD. CP predicts higher critical speed than CV, and the threshold for AO is in between them. However, AD can be approximated by CP If So Is larger than 100 (a heavily loaded bearing), on the other hand, AD can be approximated by CV if So Is less than 1 (a lightly loaded bearing) where the,bearing number So Is defined as So= PNr /(p c ). 

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