The stability condition

Many finite difference formulae have the undesirable property that small initial and rounding errors become larger as the calculation proceeds, which in the end produces a false result. This phenomenon is called (numerical) instability. In contrast a difference formula is stable when the errors become smaller during the calculation run and therefore their effect on the result declines. Most difference equations are only conditionally stable, that is they are stable for certain step or mesh sizes. The explicit equation (2.240) belongs to this group, as it is only stable when the modulus M satisfies the condition 

When Ax is given, the size of the time step At cannot be chosen at will, because violation of (2.242) will not just make the process inaccurate, it will be rendered 

The stability condition (2.242) can be derived in a number of ways cf. the extensive discussion in [2.57]. A general condition for stability of explicit difference formulae is the requirement that no coefficient in such an equation is negative, cf. [2.60]. This means for (2.240) 

The finite difference formula with M = 1/2 is just about stable, the error is shared between two neighbouring points and slowly declines. In addition the grid divides into two grid sections which are not connected, as in the difference formula 

An error becomes greater with every time step, such that the solution of the difference equation does not correspond to the solution of the differential equation as the stability condition has been violated. 

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The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

Substituting (2.33) and (2.28) into the stability condition (2.32) yields the inequality... 

The left-hand side of the inequality is the slope of the Rayleigh line, and the right-hand side is the slope of the isentrope centered on the initial state. We showed in Section 2.5 that the isentrope and Hugoniot are tangent at the initial state. Thus, the stability condition which requires that the shock wave be supersonic with respect to the material ahead of it is equivalent to the statement that the Rayleigh line must be steeper than the Hugoniot at the initial state. 

The stability condition that the shock wave is subsonic with respect to the shocked material behind it is equivalent to the statement that the Hugoniot must be steeper than the Rayleigh line at the final state. 

The consequences of the stability condition are clearly demonstrated by considering the univariant equilibrium... 

Extensive experiments were in fact needed before optimal test and acquisition conditions were eventually set (for details, see ). In any fixed strain and frequency conditions, data acquisition is made in order to record 10,240 points at the rate of 512 pt/s. Twenty cycles are consequently recorded at each strain step, with the immediate requirement that the instrument is set in order to apply a sufficient number of cycles (for instance, 40 cycles at 1.0 Hz, 20 cycles at 0.5 Hz the stability condition with the RPA) for the steady harmonic regime to be reached. Data acquisition is activated as the set strain is reached and stable. 

We give a brief survey afforded by the above results scheme (II) converges uniformly with the same rate as in the grid L2(w )-norm (see (35)) if and only if condition (39) holds. The stability condition (39) in the space C for the explicit scheme with cr = 0, namely r < coincides with the... 

With the notation h = min the stability condition (9) takes the form... 

For the purposes of the present section, let us estimate a solution of problem (4b) in terms of the right-hand side ip, provided the stability condition (21) holds. Having stipulated this condition, estimate (20) is certainly true for a solution of problem (23) and takes for now the form... 

It is easy to verify that the stability condition in the space of the two-layer scheme (29)... 

What is more, the operators A and D are commuting AD = DA. Due to these properties the stability condition for scheme (74) is expressed by... 

If isolated, the EC remains stable for all values of q corresponding to / > 0, as can be seen from Eq. (7). The major difference from the elastic dimer is that electrical energy of an isolated EC is positive and finite for all q and / while for the dimer it goes to —oo when / 0. In the next section we reexamine this result and show that the uniformity assumption for the charge distribution a across the plates of a capacitor can fail, and this can influence the stability condition for an isolated EC. 

The stability conditions are selected to maximize (C) in Equation 5-41. This requires dispersion coefficients of minimum value. From Figure 5-12 the lowest value of either dispersion coefficient occurs with F stability conditions. This is for nighttime conditions with thin to light... 

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

The complexity of the stability conditions increases the lower the symmetry of the crystal. For an isotropic condensed phase, such as a liquid or fluid the criteria can be simplified. Here, - C12 = 2C44 and the stability conditions reduce to... 

Equation (2) contains the value of actual dimensional bond characteristic of the given atom in the structure. In crystals with basic ionic bond, the ion radius can be applied as such dimensional bond characteristic (with a certain approximation), i.e. the stabilization condition for such structures is as follows ... 

A quantum-mechanical interpretation of Miedema s parameters has already been proposed by Chelikowsky and Phillips (1978). Extensions of the model to complex alloy systems have been considered. As an interesting application we may mention the discussion on the stabilities of ternary compounds presented by de Boer et al. (1988). In the case of the Heusler-type alloys XY2Z, for instance, the stability conditions with respect to mechanical mixtures of the same nominal composition (XY2+Z, X+Y2Z, XY+YZ, etc.) have been systematically examined and presented by means of diagrams. The Miedema s parameters, A t>, A ws1/3, moreover, have been used as variables for the construction of structural maps of intermetallic phases (Zunger 1981, Rajasekharan and Girgis 1983). 

Position C does not correspond to the lowest minimum of the energy following a small displacement, the block will return to the initial position whereas large displacements will move the block to the more stable position A. In A there is an (absolutely) stable equilibrium and in C a metastable equilibrium. For this mechanical system the stability conditions and the trends of spontaneous (natural) processes are related to minima (relative or absolute) of the gravitational potential energy. 

Such a resonator can be realized with an open cavity consisting of two plane or curved mirrors, as represented in Figure 2.7 linear cavity). Details of the stability conditions for different types of open resonators can be found elsewhere (Siegman, 1986). Other more sophisticated configurations, such as those of ring cavity lasers (Demtroder, 2003) and microlasers (Kasap, 2001) are also used. 

Prom the results presented in this chapter, it has been shown that the first step in the control problem of a CSTR should be the use of an appropriate mathematical model of the reactor. The analysis of the stability condition at the steady states is a previous consideration to obtain a linearised model for control purposes. The analysis of a CSTR linear model is carried out trough a scaling up reactor s volume in order to investigate the difference between the reactor and jacket equilibrium temperatures as the volume of the reactor changes from small to high value. 

The statistical mechanical interpretation of the stability condition is quite simple. From Eq. (2.1.3) we obtain by differentiation... 

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