Overrelaxation Method Research Articles

Quarter-sweep Modified SOR Iterative Algorithm and Cubic Spline Basis for the Solution of Second Order Two-point Boundary Value Problems

The aim of this study is to describe the formulation of Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method using cubic polynomial spline scheme for solving second order two-point linear boundary value problems. To solve the problems, a linear system will be constructed via discretization process by using cubic spline approximation equation. Then the generated linear system has been solved using the proposed QSMSOR iterative method to show the superiority over Full-Sweep Modified Successive Over-Relaxation (FSMSOR) and Half-Sweep Modified Successive Over-Relaxation (HSMSOR) methods. Computational results are provided to illustrate the effectiveness of the proposed method.

On the Convergence Regions of Generalized Accelerated Overrelaxation Method for Linear Complementarity Problems

In this paper, we use a generalized Accelerated Overrelaxation (GAOR) method and analyze the convergence of this method for solving linear complementarity problems. Furthermore, we improve on the convergence region of this method with acknowledgement of the maximum norm. A numerical example is also given, to illustrate the efficiency of our results.

Preconditioned iterative methods for solving weighted linear least squares problems

A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation (GAOR) methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the pre- conditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.

On the Successive Overrelaxation Method

Problem statement: A new variant of the Successive Overrelaxation (SO R) method for solving linear algebraic systems, the KSOR method was introduced. The treatment depends on the assumption that the current component can be used s imultaneously in the evaluation in addition to the use the most recent calculated components as in the SOR method. Approach: Using the hidden explicit characterization of linear functions to in troduce a new version of the SOR, the KSOR method. Prove the convergence and the consistency analysis of the proposed method. Test the method through application to well-known examples. Results: The proposed method had the advantage of updating the first component in the first equation from the firs t step which affected all the subsequent calculatio ns. It was proved that the KSOR can converge for all po ssible values of the relaxation parameter, ω*∈R-(- 2, 0) not only for ( ω∈(0, 2) as in the SOR method. A new eigenvalue funct ional relation similar to that of the SOR method between the eigenvalues of the it eration matrices of the Jacobi and the KSOR methods was proved. Numerical examples illustrating this treatment, comparison with the SOR with optimal values of the relaxation parameter were con sidered. Conclusion: The relaxation parameter ω* in the proposed method, can take values, ω*∈R-(-2, 0) not only for ( ω∈(0, 2) as in the SOR. The enlargement of the domain has the affect of relaxin g the sensivity near the optimum value of the relaxation parameter. Moreover, all the advantages of the SOR method are conserved and the proposed method can be applied to any system. This approach is promising and will help in the numerical treatment of boundary value problems. Other extensions and applications for further work are mentioned .

Mean-field density functional theory of a three-phase contact line

A three-phase contact line in a three-phase fluid system is modeled by a mean-field density functional theory. We use a variational approach to find the Euler-Lagrange equations. Analytic solutions are obtained in the two-phase regions at large distances from the contact line. We employ a triangular grid and use a successive overrelaxation method to find numerical solutions in the entire domain for the special case of equal interfacial tensions for the two-phase interfaces. We use the Kerins-Boiteux formula to obtain a line tension associated with the contact line. This line tension turns out to be negative. We associate line adsorption with the change of line tension as the governing potentials change.

Geo/Geo/1/N Queue with In-Immune and Immune Service Killing Discipline

The present investigation studies a discrete time single server queue with both positive and negative arrival streams in accordance with removal of the customer from the end (RCE)-in immune and immune service killing policy. This study is a generalization of the queue with negative customers, wherein only positive customers need a service and negative customers arriving to the system can kill the already present positive customers from any where in the queue, otherwise get lost. The concept of both in-immune and immune service killing are taken into consideration. According to the in-immune killing policy, the negative customer is allowed to kill the most recent positive customer inspite of whether it is in service or not, while the immune service killing discipline suggests that the customer currently being served is immune from killing by the negative arrival. We analyze a queue with geometric arrivals of both positive and negative customers for a finite capacity system. The stationary probability distribution and other performance measures are derived in terms of the generating functions. The results so obtained are validated by the numerical method based on successive over relaxation method (SOR). We have also employed the neurro fuzzy approach for exhibiting the approximate results for various performance measures.

On product-type generalized block AOR method for augmented linear systems

The generalized inexact accelerated overrelaxation ( GIAOR) method was presented by Bai, Parlett and Wang (Numer. Math. 102(2005)1-38) for solving the augmented system of linear equations. In this paper, a product-type generalized block AOR ( PGBAOR ) method is proposed, which is a two-step generalization of the GIAOR method. Both convergence and semi-convergence of the PGBAOR method are proved for the nonsingular and the singular augmented linear systems.

A Preconditioning Algorithm for the Positive Solution of Fully Fuzzy Linear System

A linear system of equations is called a fully fuzzy linear system (FFLS) if all the quantities of this system are fuzzy numbers. We consider the positive solution of FFLS, where the modal value (center) matrix is positive definite and we develop a new approximate procedure based on preconditioning. We observe from the numerical results that our method is more accurate than the iterative Jacobi, Gauss-Seidel and Successive Over-Relaxation (SOR) methods when finding approximate solutions of FFLS.

Comparison Results on Preconditioned GAOR Methods for Weighted Linear Least Squares Problems

We present preconditioned generalized accelerated overrelaxation methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we give a numerical example to confirm our theoretical results.

Convergence acceleration in the polarization method for nonlinear periodic fields

PurposeThe purpose of this paper is to present three novel techniques aimed at increasing the efficiency of the polarization fixed point method for the solution of nonlinear periodic field problems.Design/methodology/approachFirstly, the characteristic B‐M resulting from the constitutive relation B‐H is replaced by a relation between the components of the harmonics of the vectors B and M. Secondly, a dynamic overrelaxation method is implemented for the convergence acceleration of the iterative process involved. Thirdly, a modified dynamic overrelaxation method is proposed, where only the relation B‐M between the magnitudes of the field vectors is used.FindingsBy approximating the actual characteristic B‐M by the relation between the components of the harmonics of the vectors B and M, the amount of computation required for the field analysis is substantially reduced. The rate of convergence of the iterative process is increased by implementing the proposed dynamic overrelaxation technique, with the convergence being further accelerated by applying the modified dynamic overrelaxation presented. The memory space is also well reduced with respect to existent methods and accurate results for nonlinear fields in a real world structure are obtained utilizing a normal size processor notebook in a time of about one‐half of one minute when no induced currents are considered and of about one minute when eddy currents induced in solid ferromagnetic parts are also fully analyzed.Originality/valueThe originality of the novel techniques presented in the paper consists in the drastic approximations proposed for the material characteristics of the nonlinear ferromagnetic media in the analysis of periodic electromagnetic fields. These techniques are highly efficient and yield accurate numerical results.

Heat and mass transfer from annular fins of different cross-sectional area. Part I. Temperature distribution and fin efficiency

A numerical analysis is carried out to study efficiency and temperature distribution of annular fins of different fin profiles (constant and variable cross-sectional area) when subjected to simultaneous heat and mass transfer mechanisms. The temperature and humidity ratio differences are driving forces for heat and mass transfer, respectively. Actual psychrometric relations are used in the present work instead of a linear model between humidity ratio and temperature that has been used in the literature. A non-linear model representing heat and mass transfer mechanisms was solved using a finite difference successive over-relaxation method. Solutions are obtained for temperature distribution over the fin surface in addition to fin efficiency for both fully wet and partially wet fin surfaces. The numerical results are compared with those of previous studies. It was found that one of the linear models for the relation between the humidity ratio and temperature is a reasonable approximation.

Analysis of bearing lubrication under dynamic loading considering micropolar and cavitating effects

Numerical computation of the dynamically loaded journal bearings lubricated with micropolar fluids is undertaken based on the improved Elord cavitation algorithm and over-relaxation method. The results show that the average inflow and average outflow based on the mass conservation boundary conditions are almost equal, which is in accordance with the fact. However, under the Reynolds boundary conditions, large difference is shown between the average inflow and outflow. It is also demonstrated that with the micropolar fluids lubrication, the minimum film thickness, bearing capacity and friction power loss are increased while the maximum film pressure is decreased.

Comparison of numerical and analytical approximations of the early exercise boundary of American put options

We present qualitative and quantitative comparisons of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of American put options paying zero dividends. We analyse the asymptotic behaviour of these methods close to expiration, and introduce a new numerical scheme for computing the early exercise boundary. Our local iterative numerical scheme is based on a solution to a nonlinear integral equation. We compare numerical results obtained by the new method to those of the projected successive over-relaxation method and the analytical approximation formula recently derived by Zhu [‘A new analytical approximation formula for the optimal exercise boundary of American put options’, Int. J. Theor. Appl. Finance 9 (2006) 1141–1177]. doi:10.1017/S1446181110000854

Simulation on Surface Tracking Pattern using the Dielectric Breakdown Model

The tracking pattern formed on the dielectric surface due to a surface electrical discharge exhibits fractal structure. In order to quantitatively investigate the fractal characteristics of the surface tracking pattern, the dielectric breakdown model has been employed to numerically generate the surface tracking pattern. In dielectric breakdown model, the pattern growth is determined stochastically by a probability function depending on the local electric potential difference. For the computation of the electric potential for all points of the lattice, a two-dimensional discrete Laplace equation is solved by mean of the successive over-relaxation method combined to the Gauss-Seidel method. The box counting method has been used to calculate the fractal dimensions of the simulated patterns with various exponent η and breakdown voltage O b . As a result of the simulation, it is found that the fractal nature of the surface tracking pattern depends strongly on η and O b .

Preconditioned modified AOR method for systems of linear equations

In this paper, we present three kinds of preconditioners for preconditioned modified accelerated over-relaxation (PMAOR) method to solve systems of linear equations. We show that the spectral radius of iteration matrix for the PMAOR method is smaller than that for modified accelerated over-relaxation (MAOR) method including their comparison. The comparison results show that the convergence rate of the PMAOR method is better than the rate of the MAOR method. We provide some numerical results for the evidence of our method. Also we compare our numerical results with solutions by variational iteration method. Copyright © 2009 John Wiley & Sons, Ltd.

A modified symmetric successive overrelaxation method for augmented systems

In this paper, we establish a modified symmetric successive overrelaxation (MSSOR) method, to solve augmented systems of linear equations, which uses two relaxation parameters. This method is an extension of the symmetric SOR (SSOR) iterative method. The convergence of the MSSOR method for augmented systems is studied. Numerical examples show that the new method is an efficient method.

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

Error Control of Iterative Linear Solvers for Integrated Groundwater Models

An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient method or Generalized Minimum RESidual (GMRES) method, is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models, which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of "forward error bound estimation" to explain the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed by the US Geological Survey and the California State Department of Water Resources, we observe that this error bound guides the choice of a practical measure for controlling the error in linear systems. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive Over-Relaxation (SOR) method, the most widely known iterative solver for nonsymmetric coefficient matrices. With forward error control, GMRES can easily replace the SOR method in legacy groundwater modeling packages, resulting in the overall simulation speedups as large as 7.74×. This research is expected to broadly impact groundwater modelers through the demonstration of a practical and general approach for setting the residual tolerance in line with the solution error tolerance and presentation of GMRES performance benchmarking results.

Broyden’s method for the solution of the multilevel non-LTE radiation transfer problem

This study concerns the fast and accurate solution of multilevel non-LTE radiation transfer problems. We propose and evaluate an alternative iterative scheme to the classical MALI method. Our study is indeed based on the application of Broyden's method for the solution of nonlinear systems of equations. Comparative tests, in 1D plane-parallel geometry, between the popular MALI method and our alternative method are discussed. The Broyden method is typically 4.5 times faster than MALI. It makes it also fairly competitive with Gauss-Seidel and Successive Over-Relaxation methods developed after MALI.