Overrelaxation Method Research Articles

On semi-convergence of parameterized Uzawa methods for singular saddle point problems

Bai et al. recently proposed an efficient parameterized Uzawa method for solving the nonsingular saddle point problems [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters. The optimal iteration parameters and the corresponding optimal semi-convergence factor for the parameterized Uzawa method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the parameterized Uzawa method for solving singular saddle point problems.

Transient Double-Diffusive Convection of Water Around 4°C in a Porous Cavity

The buoyancy-driven transient double-diffusive convection in a square cavity filled with water-saturated porous medium is studied numerically. While the right and left side wall temperatures vary linearly from θa to θo and θo to θb, respectively, with height, the top and bottom walls of the cavity are thermally insulated. The species concentration levels at the right and left walls are c1 and c2, respectively, with c1>c2. The Brinkman–Forchheimer extended Darcy model is considered to investigate the average heat and mass transfer rates and to study the effects of maximum density, the Grashof number, the Schmidt number, porosity, and the Darcy number on buoyancy-induced flow and heat transfer. The finite volume method with power law scheme for convection and diffusion terms is used to discretize the governing equations for momentum, energy, and concentration, which are solved by Gauss–Seidel and successive over-relaxation methods. The heat and mass transfer in the steady-state are discussed for various physical conditions. For the first time in the literature, the study of transition from stationary to steady-state shows the existence of an overshooting between the two cells and in the average Nusselt number. The results obtained in the steady-state regime are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof number, Schmidt number, porosity and Darcy number, and midheight velocity profiles. It is found that the effect of maximum density is to slow down the natural convection and reduce the average heat transfer and species diffusion. The strength of convection and heat transfer rate becomes weak due to more flow restriction in the porous medium for small porosity.

Natural Convection in a Cavity with Orthogonal Heat-Generating Baffles of Different Lengths

, A. K. ABDUL HAKEEM,and P. KANDASWAMYUGC-DRS Center for Fluid Dynamics, Department of Mathematics,Bharathiar University, Coimbatore 641 046, INDIANatural convection in a closed square cavity induced by two mutu-ally orthogonal heat-generating baffles is considered. A numerical in-vestigation is made to understand the effect of baffle dimensions onthe resulting heat transfer characteristics. The coupled nonlinear gov-erning equations were solved by the finite difference method usingthe Alternating Direction Implicit technique and the Successive Over-Relaxation method. The results obtained indicate that the flow andtemperature fields strongly depend on the dimensions of heat-generat-ing baffles. It is found that when both baffles are mounted inside thecavity an increase in the length of any of the baffle results in a pro-portionate increase in the overall heat transfer rate. But no significantchanges in the overall heat transfer rate occur for different positionsof the baffles. When the vertical baffle is mounted on the cavity wallit extracts heat energy from the cavity and behaves like a cold wallfor an increase in its length. The study provides additional basic de-sign information during fabrication in microelectronics industry.* * *

Solution of finite arrays using stationary block iterative techniques

The finite element-boundary integral method is used in the context of stationary iteration techniques to solve for the radiation patterns of linear and planar finite arrays of identical elements. The radiating element is a cavity-backed slot/patch (CBS/P) antenna which is decomposed into an interior region and an aperture region. The array elements are coupled to each other through the aperture unknowns and the free-space Green's function. The resulting block matrix system is solved using one of three stationary block iterative techniques namely the Jacobi method, the Gauss–Seidel method and the successive over-relaxation method. An incomplete LU factorisation of the single-element matrix is used within the block iteration scheme. The rate of convergence and computational statistics of these block iterative techniques are examined and compared with each other. Conclusions are drawn on the effectiveness of each of these iterative techniques to tackle moderate to large finite arrays of CBS/P antennas.

A modified SSOR iterative method for augmented systems

Golub, Wu and Yuan [G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] have presented the SOR-like algorithm to solve augmented systems. In this paper, we present the modified symmetric successive overrelaxation (MSSOR) method for solving augmented systems, which is based on Darvishi and Hessari’s work above. We derive its convergence under suitable restrictions on the iteration parameter, determine its optimal iteration parameter and the corresponding optimal convergence factor under certain conditions. Finally, we apply the MSSOR method to solve augmented systems.

Valuation of American options by the gradient projection method

We study an equivalent optimization problem with an inequality constraint and boundary conditions, whose necessary condition for optimality is the variational inequality presentation of American options. To solve the problem, we use the gradient projection method, with discretizations both in time and space. We tested the algorithm and compared with the projective successive over-relaxation method.

Rotated Krylov preconditioned iterative schemes in the solution of convection–diffusion equations

This paper presents the numerical solution of the two-dimensional convection–diffusion partial differential equation (pde) discretized by several group iterative schemes based on the centred and rotated (skewed) five-point finite difference discretizations, namely the explicit group (EG) and explicit decoupled group (EDG) methods, respectively [W.S. Yousif, D.J. Evans, Explicit group over-relaxation methods for solving elliptic partial differential equations, Mathematics and Computer in Simulation 28 (1986) 453–466; A.R. Abdullah, The four-point explicit decoupled group (EDG) method: a fast Poisson solver, International Journal of Computer Mathematics 38 (1991) 61–70]. The application of a modified 2 × 2 block factorization preconditioner applied to the linear systems that arise from these group iterative schemes is discussed. Several Krylov subspace methods, such as Bi-CGSTAB, GMRES and TFQMR will be used to solve the preconditioned systems. The experimental results show that the condition numbers of the coefficient matrices of the transformed preconditioned systems are substantially reduced and thus result in improved convergence rates. The comparative performance analysis between the group schemes also indicate that the preconditioned EDG with Bi-CGSTAB acceleration technique is able to gain the most efficiency amongst the schemes tested.

Optimal parameters of GSOR-like methods for solving the augmented linear systems

For the augmented system of linear equations, Golub et al. [G.H. Golub, X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] studied an SOR-like method, by further accelerating it with another parameter, Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems Numer. Math. 102 (2005) 1–38] gave out a generalized SOR method. By considering a new splitting of the coefficient matrix, this paper presents another generalization of the SOR-like method (GSOR-like) which is different from the method in the Bai et al.’s paper (2005), and mainly discusses the selection of the optimal parameters. Theoretical analyses show that the convergence region for the relaxation parameter ω in our method properly contains that of the Golub et al.’s paper (2001) and our method has the same optimal asymptotic convergence rate with the method in the Bai et al.’s paper (2005). Further, the numerical example given shows the superiority of the GSOR-like method.

The optimal relaxation parameter for the SOR method applied to the Poisson equation in any space dimensions

The finite difference discretization of the Poisson equation with Dirichlet boundary conditions leads to a large, sparse system of linear equations for the solution values at the interior mesh points. This problem is a popular and useful model problem for performance comparisons of iterative methods for the solution of linear systems. To use the successive overrelaxation (SOR) method in these comparisons, a formula for the optimal value of its relaxation parameter is needed. In standard texts, this value is only available for the case of two space dimensions, even though the model problem is also instructive in higher dimensions. This note extends the derivation of the optimal relaxation parameter to any space dimension and confirms its validity by means of test calculations in three dimensions.

Preconditioned GAOR methods for solving weighted linear least squares problems

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.

On parameterized inexact Uzawa methods for generalized saddle point problems

For the large sparse saddle point problems, Bai et al. recently studied a class of parameterized inexact Uzawa (PIU) methods [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang. On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. For a special case that the (1,1)-block is solved exactly, they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method. In this paper, we develop these methods to the large sparse generalized saddle point problems. For the obtained parameterized inexact Uzawa method, we prove its convergence under suitable restrictions on the iteration parameters. In particular, we determine its quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor for the saddle point problems. Furthermore, This PIU method is generalized to obtain a framework of accelerated variants of the parameterized inexact Uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration.

An Adaptive Successive Over-Relaxation Method for Computing the Black-Scholes Implied Volatility

A new successive over-relaxation method to compute the Black-Scholes implied volatility is introduced. Properties of the new method are fully analyzed, including the well-definedness, and local and global convergence patterns. Quadratic order of convergence is achieved by either a transformation of sequence technique or dynamic relaxation. The method is further enhanced by introducing a rational approximation on initial values. Numerical implementation shows that uniformly in a very large approximation domain, the new method converges to the true implied volatility with very few iterations. Overall, the new method achieves a very good combination of efficiency, accuracy and robustness.

An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility

A new successive over-relaxation method to compute the Black–Scholes implied volatility is introduced. Properties of the new method are fully analysed, including global well-definedness, local convergence, as well as global convergence. Quadratic order of convergence is achieved by either a dynamic relaxation or transformation of sequence technique. The method is further enhanced by introducing a rational approximation on initial values. Numerical implementation shows that uniformly in a very large domain, the new method converges to the true implied volatility with very few iterations. Overall, the new method achieves a very good combination of efficiency, accuracy and robustness.

An iterative method for positive real systems

In this paper we present a general form of an iterative method to approximate the solution of the linear system A x = b, when the matrix A + AT is real and positive definite or equivalently A is positive real. Under some assumptions we generalise several results given by Li, Liang and Evans. This iterative method contains an auxiliary matrix D 1 and, for a suitable choice of the matrix D 1, we establish convergence conditions for this iterative method. As a particular case we have the point accelerated over-relaxation method (AOR) and we analyse its convergence. Some comments about the point case of this iterative method and the one presented by Li, Liang and Evans are made in the last remarks of this paper.

A Finite-Volume Solution with a Bidirectional Upwind Difference Scheme for the Three-Dimensional Radiative Transfer Equation

Abstract A new calculation scheme is proposed for the explicitly discretized solution of the three-dimensional (3D) radiation transfer equation (RTE) for inhomogeneous atmospheres. To separate the independent variables involved in the 3D RTE approach, the spherical harmonic series expansion was used to discretize the terms, depending on the direction of the radiance, and the finite-volume method was applied to discretize the terms, depending on the spatial coordinates. A bidirectional upwind difference scheme, which is a specialized scheme for the discretization of the partial differential terms in the spherical harmonic-transformed RTE, was developed to make the equation determinate. The 3D RTE can be formulated as a simultaneous linear equation, which is expressed in the form of a vector–matrix equation with a sparse matrix. The successive overrelaxation method was applied to solve this equation. Radiative transfer calculations of the solar radiation in two-dimensional cloud models have shown that this method can properly simulate the radiation field in inhomogeneous clouds. A comparison of the results obtained using this method with those using the Monte Carlo method shows reasonable agreement for the upward flux, the total downward flux, and the intensities of radiance.

Effect of baffle–cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles

Buoyancy driven convection in a square cavity induced by two mutually orthogonal and arbitrarily located baffles is studied numerically. The baffles are of different sizes and the flow is two-dimensional. The coupled governing equations were solved by finite difference method using Alternating Direction Implicit technique and Successive Over-Relaxation method. The steady state results are presented in the form of streamline and isotherm plots. It is found that buoyancy force plays a key role and overall heat transfer in the cavity is enhanced for higher values of both baffle–cavity ratios. Flow inhibition emerges as a deciding factor and diminishes heat transfer when the horizontal baffle is located above the center of the cavity.

Generalized AOR Algorithms for Linear Complementarity Problem

In this paper, we propose a class of generalized accelerated overrelaxation (GAOR) methods, in which one special case reduces to the generalized successive overrelaxation(GSOR) methods, for solving a linear complementarity problem LCP( M, q). Sufficient conditions for convergence of the GAOR and GSOR methods are presented for cases when the system matrix M is an H–matrix, M–matrix and a strictly or irreducible diagonally dominant matrix. Moreover, when M is an L–matrix, we examine the monotone convergence of the new methods. Finally a numerical example is used to validate the results proved in this paper.

Discrepancy between published Laplace difference equations on cylindrical dielectrics

The problem of two unequal electrodes within a shielding cavity in the presence of a dielectric partially filling the gap is treated by means of overrelaxation methods in the axisymmetric case. A discrepancy in the literature concerning the boundary equation between two dielectrics is resolved and the difference equation for the dielectric corner point is obtained by means of Gauss’ law. Some numerical results are described.

Natural Convection in a Square Cavity in the Presence of Heated Plate

Natural convection heat transfer in a square cavity induced by heated plate is studied numerically. Top and bottom of the cavity are adiabatic, the two vertical walls of the cavity have constant temperature lower than the plate’s temperature. The flow is assumed to be two-dimensional. The discretized equations were solved by finite difference method using Alternating Direction Implicit technique and Successive OverRelaxation method. The study was performed for different values of Grashof number ranging from 103 to 105 for different aspect ratios and position of heated plate. Air was chosen as a working fluid (Pr = 0.71). The effect of the position and aspect ratio of heated plate on heat transfer and flow were addressed. With increase of Gr heat transfer rate increased in both vertical and horizontal position of the plate. When aspect ratio of heated thin plate is decreased the heat transfer also decreases. For the vertical situation of thin plate heat transfer becomes more enhanced than for horizontal situation.

Symmetric block-SOR methods for rank-deficient least squares problems

In this article, we develop symmetric block successive overrelaxation (S-block-SOR) methods for finding the solution of the rank-deficient least squares problems. We propose an S2-block-SOR and an S3-block-SOR method for solving such problems and the convergence of these two methods is studied. The comparisons between the S2-block and the S3-block methods are presented with some numerical examples.