Successive 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. 

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.

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. 

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... 

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]. 

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. 

This program applies the methoa of Successive Over-Relaxation to the solution of an elliptic PDE. 

Mangasarian, O. L., Musicant, D. R. (1999). Successive over relaxation for support vector machines. IEEE Trans. Neural Networks 10,1032-1036. 

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

Several iterative techniques have now been discussed for approximating the solution of the set of matrix equations associated with two dimensional BVPs. The most important of these are Gauss-Seidel, simple over-relaxation, successive over relaxation (SOR) and Chebyshev (with SOR). With the BVP in Listings 12.24 or... 

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