Over-relaxation method

Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm... 

M-Shake This is Newton-iteration-based implementation of SHAKE, using (4.18)-(4.19) to solve Eqs. (4.27)-(4.29). Methods like this were first proposed by Ciccotti and Ryckaert [84] in the context of rigid body molecular dynamics. An extended discussion of such methods with reference to their convergence, implementation, in particular linear system solvers, and variants such as SHAKE-SOR (which uses the successive over-relaxation method) can be found in [25]. A conjugate gradient method can also be used [392]. 

For choices of 0 < < 1, we have under-relaxation methods, which are successful for some systems that are not convergent for Gauss-Seidel. Those methods associated with > 1 are called over-relaxation methods and are useful in accelerating the convergence for systems that are already convergent by Gauss-Seidel. These over-relaxation methods are also named successive over-relaxation (SOR), and find application in the numerical solution of certain partial differential equations. 

In the numerical solution of the Reynolds equation It was assumed that all the variables were distributed quadratlcally in the coordinate directions. Derivatives were thus expressed In terms of three adjacent points, (i,j-l), (i,j) and (l,j+l) In the radial (R) direction and (1-1,j), (l,j) and (1+1,j) in the circumferential (6) direction as shown In Figure 2. The Gauss-Seldel Iterative over-relaxation method was used to solve equation (17) written In finite difference form. 

Figure 12.62. Examples of dependency of corrective error on number of iterations for various over relaxation methods.

Figure 13. Voltage relaxation method for the determination of the diffusion coefficients (mobilities) of electrons and holes in solid electrolytes. The various possibilities for calculating the diffusion coefficients and from the behavior over short (t L2 /De ) and long (/ L2 /Dc ll ) times are indicated cc h is the concentration of the electrons and holes respectively, q is the elementary charge, k is the Boltzmann constant and T is the absolute temperature.

Eq. 16 is an extremely useful criterion for examining the extent of dipolar interaction in a multispin system, and gives the relaxation method a major advantage over the n.O.e. method. The equivalent quantitative test for the n.O.e. experiment requires all but the receptor nucleus to be saturated and this is not readily performed in practice. 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

Kao and Tsien studied the Ca +-binding kinetics of fura-2 and azo-1 by temperature-jump relaxation methods. In 140 mM KCl at 20°C, the respective association and dissociation rate constants for fura-2 were 6x10 M s and 97 s these kinetic properties were insensitive to hydrogen ion concentration over the pH range from 7.4 to 8.4. Azo-1 was studied in 140 mM KCl At 10°C, azo-... 

The advantage of the relaxation method over the iteration methods is simply that, in the experience of the authors, any process can be solved. Processes for which the iteration methods will not converge, inevitably converge by the relaxation method. Because of the surety of solution the authors wish to stress the relaxation method in this article and have presented a general program on vapor-liquid processes for the use of engineers who need solutions to separation process problems. 

In electrode kinetics, however, the charge transfer rate coefficient can be externally varied over many orders of magnitude through the electrode potential and kd can be controlled by means of hydrodynamic electrodes so separation of /eapp and kd can be achieved. Experiments under high mass transport rate at electrodes are the analogous to relaxation methods such as the stop flow method for the study of reactions in solution. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

Terzopoulos (1986) has shown how to speed up the computation using multi-grid relaxation methods. He uses a hierarchy of multi-resolution grids where data propagates upward as well as downward through the hierarchy. The use of a multi-resolution pyramid allows information to propagate faster over larger distances. 

Iterative methods (like Gauss-Seidel, Successive over relaxation and conjugate gradient) have often been preferred to the... 

Since the interface relaxation method was established it has been used to determine the coexistence conditions for over 20 polymer mixtures [74,75,88,91, 92,95-99]. 

Termination rate coefficients can be measured using the y-radiolysis relaxation method. This involves initiation using y-radiation, followed by removal of the reaction vessel from the y-source. Conversion during the relaxation period is monitored by dilatometry, and the decay in polymerization rate over time is related to the rate of radical loss. When large particles are used, radical loss is dominated by intraparticle termination, rather than exit into the aqueous phase, and the rate coefficient for termination can be determined from the decay curve. By using multiple insertions and removals, the termination rate coefficient is determined over a wide range of polymer mass fraction (wp). 

When 0=1, the original Gauss-Seidel method is recovered. Other values of the parameter a yields different iterative sequences. If 0 < a < 1 then the procedure is an under-relaxation method, else with a > 1 we have obtained an approach that is called the successive over-relaxation (SOR) technique. 

The p-jump method has several advantages over the t-jump technique. Pressure-jump measurements can be repeated at faster intervals than those with t-jump. With the latter, the solution temperature must return to its ini-lial value before another measurement can be conducted. This may take 5 min. With p-jump relaxation, one can repeat experiments every 0.5 min. One can also measure longer relaxation times with p-jump than with t-jump relax-mion. As noted earlier, one of the components of a t-jump experiment is It heat source such as Joule heating. Such high electric fields and currents can destroy solutions that contain biochemical compounds. Such problems lIo not exist with the p-jump relaxation method. 

Yield stresses of tomato concentrates from the juice evaporation process using four FSO were determined (AS.) by the stress relaxation method (18.) over the concentration range of 9 to 14% T.S. The magnitudes of yield stress increased not only with total solids content but also with the FSO. The use of larger finisher screens resulted in concentrates with higher yield stress. 

Chemical relaxation methods [52] show evidence of a distribution of relaxation frequencies rather than a single one as found with classic ion detergents [193]. Thus, in agreement with the above conclusions, NaC and NaDC apparently self-associate over a whole range of concentrations and not at some critical micellar concentration. Further, the relaxation frequencies are strongly concentration dependent, suggesting that the distribution of aggregate sizes is wide and shifts upwards as bile salt concentration is increased [52]. 

Among the attractive features of the dielectric relaxation method are its relative case of application and the availability of a very large frequency range in a more or less continuous manner. In fact, measurements can be made over the frequency range from 1CF4 Hz to 3 x 1010 Hz, using a variety of techniques. These are summarized in Table 7-1. 

The relaxation methods employed are Dynamic Mechanical Thermal Analysis (DMTA) and Dielectric Thermal Analysis (DETA). Generally in both cases a single excitation frequency is used and the temperature is varied, typically over a range between — 100 °C and +200 °C. Changes in molecular motion, and hence 7, are detected by both techniques, but in the case of DETA the process has to involve movement of dipoles or fully developed electrical charges on the polymer in order to be detected. Thus the two techniques can be used to complement each other, since transitions can be detected on DMTA and assigned as due to dipoles according to whether or not they also occur with DETA. 

Pulse methods dominate over (CW) methods in the instrumentation used in NMR spectroscopy, while the reverse is true for ESR. The problem in adopting pulse methods for the latter arises for two main reasons (1) a typical ESR spectrum of a free radical occupies 90 MHz, and (2) spin-lattice relaxation... 

Added in proof A new iterative method, called the modified Chebyshev semi-itera-live method, eliminated this factor of two in the cyclic case, and is more rapidly convergent than the successive over-relaxation iterative method. See [18a]. 

The photochemical relaxation method enqployed in this work initiates a chemical reaction by the direct absorption of light into one of the reacting species (1). In contrast, the temperature and pressure-jump techniques perturb the position of equilibrium by changing the conditions of the reaction medium (2), The photochemical dissociation method thus has several inherent advantages over the ten5>erature-jump and related relaxation methods... 

Auer and Gauss showed that for the CC calculations of the SSCCs with relaxation of the reference orbitals in the presence of the perturbations, unphysical results are obtained over a wide range of the potential curve. The effect of triplet instability in the relaxed methods was most dramatic for perturbative approaches like CCSD(T). CC calculations using the unperturbed HF orbitals (6 /0a = 0), i.e., so-called unrelaxed calculations, did not show any triplet instability. 

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