MULTICOMPONENT ITERATIVE METHODS SOLVING STATIONARY PROBLEMS OF MATHEMATICAL PHYSICS

2008 ◽  
Vol 13 (3) ◽  
pp. 313-326
Author(s):  
Natali G. Abrashina-Zhadaeva ◽  
Alexey A. Egorov
Keyword(s):  
Convergence Rate ◽  
Iterative Methods ◽  
Arbitrary Number ◽  
Mathematical Physics ◽  
Optimal Values ◽  
Stationary Problems ◽  
Number Of Iterations

Additive iterative methods of complete approximation for stationary problems of mathematical physics are proposed. The convergence rate in the case of an arbitrary number of commutative and noncommutative partition operators is analysed. The optimal values of the iterative parameter are found and related estimates for the number of iterations are derived. Some applications of suggested iterative methods are discussed.

scholarly journals Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives

2019 ◽  
Vol 55 (2) ◽  
pp. 207-215
Author(s):  
A. U. Prakonina
Keyword(s):  
Convergence Rate ◽  
Iterative Methods ◽  
Imaginary Part ◽  
Conjugate Gradient ◽  
Discrete Models ◽  
Lu Factorization ◽  
The Matrix ◽  
Mixed Derivatives ◽  
Incomplete Lu Factorization ◽  
Number Of Iterations

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

scholarly journals Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 130 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Iterative Procedure ◽  
Accurate Solution ◽  
Final Solution ◽  
Worst Case ◽  
Flow Friction ◽  
One Step ◽  
Newton Raphson ◽  
Number Of Iterations

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

scholarly journals Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration

2019 ◽  
Vol 16 (3) ◽  
pp. 1430
Author(s):  
Nur Afza Mat Ali ◽  
Rostang Rahman ◽  
Jumat Sulaiman ◽  
Khadizah Ghazali
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Iterative Method ◽  
Iterative Methods ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sparse Linear System ◽  
Successive Over Relaxation ◽  
Number Of Iterations

<p>Similarity method is used in finding the solutions of partial differential equation (PDE) in reduction to the corresponding ordinary differential equation (ODE) which are not easily integrable in terms of elementary or tabulated functions. Then, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is applied in solving the sparse linear system which is generated from the discretization process of the corresponding second order ODEs with Dirichlet boundary conditions. Basically, this ODEs has been constructed from one-dimensional reaction-diffusion equations by using wave variable transformation. Having a large-scale and sparse linear system, we conduct the performances analysis of three iterative methods such as Full-sweep Gauss-Seidel (FSGS), Full-sweep Successive Over-Relaxation (FSSOR) and HSSOR iterative methods to examine the effectiveness of their computational cost. Therefore, four examples of these problems were tested to observe the performance of the proposed iterative methods.  Throughout implementation of numerical experiments, three parameters have been considered which are number of iterations, execution time and maximum absolute error. According to the numerical results, the HSSOR method is the most efficient iterative method in solving the proposed problem with the least number of iterations and execution time followed by FSSOR and FSGS iterative methods.</p>

scholarly journals A Modified SSOR Preconditioning Strategy for Helmholtz Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Shi-Liang Wu ◽  
Cui-Xia Li
Keyword(s):  
Convergence Rate ◽  
Linear Systems ◽  
Iterative Methods ◽  
Low Cost ◽  
Computational Cost ◽  
Coefficient Matrix ◽  
Computational Time ◽  
Helmholtz Equations ◽  
Speed Up ◽  
Methods Numerical

The finite difference method discretization of Helmholtz equations usually leads to the large spare linear systems. Since the coefficient matrix is frequently indefinite, it is difficult to solve iteratively. In this paper, a modified symmetric successive overrelaxation (MSSOR) preconditioning strategy is constructed based on the coefficient matrix and employed to speed up the convergence rate of iterative methods. The idea is to increase the values of diagonal elements of the coefficient matrix to obtain better preconditioners for the original linear systems. Compared with SSOR preconditioner, MSSOR preconditioner has no additional computational cost to improve the convergence rate of iterative methods. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners.

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