A Parallel Symmetric Successive Overrelaxation Method for OVERFLOW

Chapter 15 - SYMMETRIC SOR METHOD AND RELATED METHODS

In this paper, a class of parallel blockwise matrix multisplitting block relaxation methods, including the blockwise matrix multisplitting block symmetric accelerated overrelaxation method, the blockwise matrix multisplitting block unsymmetric and symmetric successive overrelaxation methods and the blockwise matrix multisplitting block unsymmetric and symmetric Gauss-Seidel methods, etc., is established for the large sparse block system of linear equations, and its convergence theory is set up thorouthly when the coefficient matrix is a block H-matrix. Also, the new methods are further extended by relaxing different block elements of the iterations with different relaxation parameters and, therefore, general frameworks of parallel blockwise matrix multisplitting block relaxation methods for solving the block system of linear equations are naturally obtained.

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 .

In this paper, we present an efficient numerical method for solving the linear complementarity problems based on preconditioning strategy. Furthermore, the convergence properties of the proposed method have been analyzed and compared with symmetric successive over-relaxation (SSOR) method. Finally, some numerical experiments are illustrated to show the efficiency of the proposed method.

We study symmetric successive overrelaxation (SSOR) method for absolute complementarity problems. Solving this problem is equivalent to solving the absolute value equations. Some examples are given to show the implementation and efficiency of the method.

In this paper we consider the solution of large linear systems with Hermitian coefficient matrix. In particular, we find sufficient conditions for convergence of the Symmetric Successive Overrelaxation (SSOR) method.

CHAPTER 5 - An Adaptive Chebyshev Procedure Using Special Norms

p-cyclic matrices and the symmetric successive overrelaxation method

In this paper we consider various iterative methods for solving systems of linear algebraic equations. We shall be primarily concerned with large systems with sparse matrices such as arise in the solution of elliptic boundary value problems in two dimensions by finite difference methods. Our discussion is divided into two parts: First, we review the well-known facts about such methods as the Jacobi, Gauss-Seidel, and successive overrelaxation methods. A treatment of various acceleration techniques is included. In the second part of the paper we consider the symmetric overrelaxation method and describe some practical procedures which can be used to obtain very rapid convergence. By showing that the method can be very effective in many cases and by outlining a definite procedure for its application, we hope to encourage its wider usage and to stimulate further research.

The application of parallel computers to reservoir simulation is studied with the aid of two recently introduced Multiple Instruction, Multiple Data (MIMD) computers. The problems posed by reservoir simulators are shown to be particularly well suited for such parallel computers* Fundamental parallel programming techniques using Fortran are presented for the Hetrogeneous Element Processor (HEP) produced by Denelcor and the iPSC Hypercube produced by Intel. These techniques are then applied to two of the most time consuming tasks in reservoir simulation: forming the matrix coefficients and producing a sparse matrix solution. The problem of parallel formation of sparse matrix coefficients is addressed for the single and multiphase cases using both black oil and compositional fluid models. Parallel sparse matrix solution is addressed by considering D4 ordered Gaussian elimination, and the multigrid, conjugate gradient, and Successive Over-Relaxation (SOR) methods. Three parallel algorithms based on SOR are developed to illustrate the possible diversity of parallel algorithms. The red-black and multicolored SOR methods which are often used on vector machines are shown to be easily adaptable to parallel MIMD machine. Matrix partitioning and a new method of iteration pipelining are also presented as parallel SOR methods. The various SOR algorithms, in parallel form, are mapped onto the HEP to illustrate the speedups currently possible on an MIMD machine. Projections are made on the future importance of these types of machines to reservoir simulation.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

Successive over-relaxation SOR is a computationally intensive, yet extremely important iterative solver for solving linear systems. Due to recent trends of exponential growth in the amount of data generated and increasing problem sizes, serial platforms have proved to be insufficient in providing the required computational power. In this paper, we present parallel implementations of red-black SOR method using three modern programming languages namely Chapel, D and Go. We employ SOR method for solving 2D steady-state heat conduction problem. We discuss the optimisations incorporated and the features of these languages which are crucial for improving the program performance. Experiments have been performed using two, four, and eight threads and performance results are compared with those obtained using serial execution. The analysis of results provides important insights into the working of SOR method.

Accelerated generalized successive overrelaxation method for least squares problems

This paper is concerned with the numerical solution of the secondary vorticity equations in curved ducts, cascades, and rotors. The classical approach of splitting the flow into primary and secondary flow fields is employed and extended to include effects of viscosity, rotation, and density stratification. All elliptical effects are neglected and the Crank-Nicolson method is used to solve the secondary vorticity equation. The secondary flow field is obtained by solving a Poisson equation using a successive over-relaxation method. The results are compared with theoretical and experimental data from stationary ducts, compressor and turbine cascades. The agreement is good for most of the cases.

CHAPTER 12 - The Nonsymmetrizable Case