Over relaxation

Nicholls, A., Honig, B. A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation. J. Comp. Chem. 12 (1991) 435-445. 

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... 

There has been a debate at the European level over relaxing DTC advertising in a limited number of therapeutic areas. At the time of writing this chapter these suggested changes have been shelved and there are no plans to allow even limited DTC within the EU. 

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. 

Figure 10.12 Time to convergence computed for different values of p. Convergence is considerably faster if successive over-relaxation is used.

When uj < 1, we have so-called under-relaxation technique, often used with nonlinear problems. For example, when solving for the non-linear velocity distribution using a shear thinning power law model, the fastest solution is achieved when to = n since n < 1. When to > 1, SOR becomes an over-relaxation technique. 

Figure 8.14 Convergence for the SOR iterative solution scheme with various over-relaxation parameters for the FD compression molding problem.

There are, however, some side effects of epidural anesthesia. The patient s legs often become numb, and movement of the leg muscles is difficult. Other mild side effects include a backache (thought to be a result of over-relaxation of the back muscles), itchiness of the skin, and an inability to urinate. More rarely, patients experience headache, lowered blood... 

Algorithm, Utilizing Successive Over-Relaxation to Solve the Poisson-Boltzmann Equation. 

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

Nicholls A, Honig B (1991) Rapid finite difference alogrithm, utililizing successive over-relaxation to solve the Poisson-Boltzman equation, J Comput Chem, 12 435-445... 

A. Nicholls and B. Honig, /. Comput. Chem., 12, 435 (1991). A Rapid Finite Difference Algorithm, Utilizing Successive Over-Relaxation to Solve the Poisson-Boltzmann Equation. 

The equations (15a-c) are discretized with upwind finite differences and the resulting linear systems of equations are solved using the Succesive Over Relaxation (SOR) technique. 

Reaction rate oscillations may be accompanied by temperature oscillations [temperature fluctuations of up to 500 K have been reported (24)] or they may be isothermal. Isothermality occurs either because the catalyst can conduct heat away much faster than the rate at which it is produced by the reaction, as is the case in UHV studies, or because isothermal conditions are forced on the system by anemometry, as described in the work of Luss and co-workers (757). Oscillation frequencies can range from more than 10 Hz (24) up to periods of several hours (217,219). Often there is evidence for several time scales in a single oscillating stem. Relatively regular high-frequency oscillations may be superimposed over relaxation oscillations (93,98), with the two types of oscillations caused by different changes on the catalyst surface. 

Combine these new values with the assumed set using the successive over-relaxation technique (4), and repeat from step 2 to convergence. 

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. 

SOM Second Order Moments SOR Successive Over-Relaxation SUPERBEE SUPERBEE function TDM A Tri-Diagonal Matrix Algorithm TVD Total Variation Diminishing UDS Upstream Differencing Scheme... 

FIGURE 4.1 First-order reaction of the type A. B. (a) Concentration relative to the original one as a function of time (t) over relaxation time (r). (b) Example of log concentration versus time tan a — k log e — 0.434C. D is the decimal reduction time. 

Eg is an activation energy for cross-over (relaxation and dissipation of energy to the lattice) from excited to ground state. It is easy to show that energy transferred from the excited state to the lattice has a certain probability. Pi, to occur, namely-... 

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]. 

Poisson-Boltzmann equation (PBE), multigrid (MG), algebraic multigrid (AMG), finite difference (FD), finite element (FE), Gauss-Seidel (GS), conjugate gradient (CG), successive over relaxation (SOR), stochastic dynamics (SD), quantum mechanics (QM), molecular mechanics (MM), molecular dynamics u, (MD). 

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]. 

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