Successive Over-Relaxation method

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 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. 

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... 

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. 

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

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. 

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]. 

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. 

The geometry and mesh arrangement in the fluid region are exactly the same as those of the steady-state subchannel analysis code. Figure 6.60 shows the entire algorithm. The momentum conservation equations for three directions and a mass conservation equation are solved with the Simplified Marker And Cell (SMAC) method [32]. In the SMAC method, a temporary velocity field is calculated, the Poison equation is solved, and then the velocity and pressure fields are calculated as shown in Fig. 6.61. The Successive Over-Relaxation (SOR) method is used to solve a matrix. 

The Jacobi, Gauss-Seidel, ancJ successive over-relaxation (SOR) methods... 

Aniansson and Wall (A-W) appear to be the first to develop a relatively more accurate and convincing kinetic model for micellization in conjunction with the multiple-equilibrium reaction scheme as shown by Equation 1.20. - The superiority of the A-W model over the others is that it predicts the presence of two discrete relaxation times (Xj and Xj) during the course of micelle formation in the aqueous solutions of a single surfactant above CMC — a fact revealed by many experimental observations in related studies. Although this model successfully predicts the presence of two discrete relaxation times, it is not fully tested in terms of (1) reproducibility of kinetic parameters derived from this model by using various chemical relaxation methods, and (2) kinetic parameters obtained from both relaxation times x, and Xj have reasonably acceptable values. 

Often the information on NMR relaxation parameters carried by image contrast is insufficient to address a particular problem. We can then look to the rich information content of the spectrum itself. Generally, spectroscopy of the entire body is not of much value, and in vivo spectroscopy is usually carried out as localized spectroscopy, that is, over a part of the body. There are various ways of restricting the operation of the spectrometer to a particular region, and they fall into two broad classes those that depend on the physical dimensions of the rf coil and those that use field gradients in the pulse sequences. Often these approaches are combined. At this time, the use of spectroscopic examinations has not become part of the repertoire of clinical practice, despite a history of in vivo spectroscopy almost as old as MRI itself. In vivo spectroscopy has had a number of landmark successes in solving problems in metabolism research in both animals and humans, but there have been no spectroscopic applications that have been demonstrated to be more effective than other methods for the routine diagnosis of disease. 

Unfortunately, the Debye model provides only an approximate description of aprotic solvents. It has been applied extensively to determine their relaxation properties quite successfully, mainly because permittivity data are available over a limited frequency range. As a result, the high-frequency parameter is usually obtained by a long extrapolation. As experimental methods have become available at frequencies above 50 GHz, it has been found that the behavior of aprotic solvents is more complex [9]. 

The pulses applied in FT NMR are phase coherent in two ways. First, the Bi field is phase coherent over the whole sample, and second, successive pulses may be applied with a defined phase relationship. Consequently, it is possible to manipulate the nuclear magnetisations by a sequence of pulses to produce different overall responses. A wide variety of pulse techniques has been developed which have greatly increased the power and versatility of NMR as an analytical method. These methods are conveniently divided into two classes, one-dimensional techniques in which a spectrum is recorded as a function of one frequency as in standard NMR, and two-dimensional techniques in which the spectrum is recorded as a function of two frequencies. Here, attention is focused on the basic types of experiment, and on the form of the spectra and the type of information obtained, rather than on the theory and operation of the techniques. Methods for determining relaxation times are described in chapters 4, 6 and 7, so are not dealt with here. 

---