Unconditional stability

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

Temptation has been strong, and not always resisted, to approach total compensation by damping the oscillations out with an appropriately placed capacitance so as to reach the ideal situation of total compensation with unconditional stability. In fact, the cure is worse than the disease. The additional response time accompanying introduction of the damping capacitance will indeed distort the Faradaic current in a more severe and undecipherable manner than does ohmic drop. 

Implicit method This method is slightly more complicated, but it offers unconditional stability. Let... 

Stability of this two-dimensional method is unconditional Stability of fully implicit method is also unconditional Only tridiagonal matrices to be solved for problems with two dimensions Usually true only in problems with one dimension, not two dimensions. 

This result is proven by making use of an energy method. We refer to [47] for a proof and for other related results, e.g., sufficient conditions on e and k for unconditional stability in L. These results have been extended by Le Meur [50] to the case of multifluid flows. 

Bou-Rabee, N., Vanden-Eijnden, E. A patch that imparts unconditional stability to explicit integrators for Langevin-hke equations. J. Comput. Phys. 231, 2565-2580 (2012). doi 10. 1016/j.jcp.2011.12.007... 

The IE scheme is nonconservative, with the damping both frequency and timestep dependent [42, 43]. However, IE is unconditionally stable or A-stable, i.e., the stability domain of the model problem y t) = qy t), where q is a complex number (exact solution y t) = exp(gt)), is the set of all qAt satisfying Re (qAt) < 0, or the left-half of the complex plane. The discussion of IE here is only for future reference, since the application of the scheme is faulty for biomolecules. 

The scheme of second-order accuracy (unconditionally stable in the asymptotic sense). Before taking up the general case, our starting point is the existing scheme of order 2 for the heat conduction equation possessing the unconditional asymptotic stability and having the form... 

It seems clear that Green s formulas are certainly true in the case when the operator A is defined in such a way. Moreover, A = A > 0. All this provides the sufficient background for the possible applications of the general stability theory outlined in Chapter 6, within the framework of which the scheme concerned is unconditionally stable for a > 0.5. 

It is clear from this expression that the method is stable for all A. and all time steps hn that is, the method is unconditionally stable (for linear problems). A consequence of the strong stability is that the time step can be chosen primarily to maintain accuracy. In the slowly varying regions of stiff problems, the time steps can be very large compared those required to maintain stability for an explicit algorithm. 

Associated with numerical problems is the concept of stability. A numerical scheme is stable when a solution is reached even with large time-steps (unsteady problems) or iteration steps (algebraic system of equations iteratively solved). Therefore the size of the time-step or of the iteration-step is dictated by stability requirements. It must be kept in mind that stability does not mean accuracy an implicit scheme of a dynamic problem is unconditionally stable but the solution obtained with large values of the time step may not be realistic. 

The main advantage of the ADI method is that the stability of this two-dimensional method is unconditional, as is the fully implicit method. In addition, each of Eqs. (10.51) and (10.52) are only... 

Another factor affecting stability is a derivative boundary condition. Keast and Mitchell pointed out potential problems with CN in this regard [334], and investigations in the electrochemical context revealed some problems with methods otherwise thought to be unconditionally stable, such as CN and Saul yev [116,117,118]. The CN method was found to become... 

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... 

For physically realistic and bounded results, it is necessary to ensure that all the coefficients of the discretization equation are positive. This requirement imposes restrictions on the time step that can be used with different values of 0. It can be seen that a fully implicit method with 0 equal to unity is unconditionally stable. Detailed stability analysis is rather complex when both convection and diffusion are present. In general, simplified criteria may be used when an explicit method is used in practical simulations ... 

Thus, for two free surfaces, the eigenvalue problem for a is reduced to finding a nontrival solution of Eq. (12-199), subject to the six boundary conditions, (12-200), (12-203), and (12-204). In particular, let us suppose that we specify Pr, Gr, and a2 (the wave number of the normal model of perturbation). There is then a single eigenvalue for a such that / / 0. If Real(er) < 0 for all a, the system is stable to infinitesimal disturbances. On the other hand, if Real(er) > 0 for any a, it is unconditionally unstable. Stated in another way, the preceding statements imply that for any Pr there will be a certain value of Gr such that all disturbances of any a decay. The largest such value of Gr is called the critical value for linear stability. 

Since all differences in Eq. (4.73) are positive according to Fig. 4.20,3 this equation is satisfied under all circumstances. Therefore, it is unconditionally stable. We obtain this stability, however, at the cost of a new algebraic complexity. Recall Eq. (4.50), in which all temperatures except T[ +l are known, and the latter is obtained by solution of the equation. By contrast, in Eq. (4.72) only T is known, and the application of this equation to the nodes yields a tridiagonal matrix which then can be solved by the method discussed in Ex. 4.1. 

Now, even if values of c" i = 1,2,..., ) are known, (25.95) cannot be solved explicitly, as c"+l is also a function of the unknown c, 1 and c" /. However, all k equations of the form (25.95) for i = 1,2,..., k form a system of linear algebraic equations with k unknowns, namely, c"+l, c"+l,..., c"+l. This system can be solved and the solution can be advanced from r to tn+. This is an example of an implicit finite difference method. In general, implicit techniques have better stability properties than explicit methods. They are often unconditionally stable and any choice of Ar and Ax may be used (the choice is ultimately based on accuracy considerations alone). 

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