Download or read book A Comparison of Iterative Methods for the Solution of Elliptic Partial Differential Equations Particularly the Neutron Diffusion Equation written by Kevin N. Schwinkendorf and published by . This book was released on 1983 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two new concepts have been explored in solving the neutron diffusion equation in one and two dimensions. At the present time, the diffusion equation is solved using source iterations. These iterations are performed in a mathematical form which has a great deal of physical significance. Specifically, the neutron production term is on the right-hand side, while the absorption and leakage terms are on the left side. In performing a single source iteration, a distribution for the neutron flux is assumed so that the production term can be calculated. This provides a "known" right-hand side. Solving the difference equation for the flux, which corresponds to this assumed source distribution, gives the next estimate for the flux distribution. This type of iteration has the physically significant characteristic of finding directly, for each iteration, a flux which corresponds to an assumed source distribution. In this thesis it was found that by subtracting the absorption term from both sides of the diffusion equation, and performing "source iterations" with both absorption and production terms on the right-hand side (and only the leakage term on the left-hand side), improved convergence rates were attained in many cases. In one neutron energy group, this new idea of putting the absorption term on the right-hand side worked best with only one region, and where reactor dimensions were large compared to the thermal neutron diffusion length (a”L). In small reactors, where a=L, convergence behavior was similar for both forms of iteration. This new idea was also found to work quite well in one-group multiregion problems. However, due to problems with numerics (inherent asymmetric treatment of the scattering terms), the method does not work at all in a multi-energy group formulation. Secondly, in two dimensions, a closed-form solution to a single source iteration has been found. At this time, the standard method of solution for a two-dimensional source iteration is to perform "inner iterations" to approximately solve for the flux that corresponds to an assumed source. The alternative, up until now, was to solve a giant matrix of the order (N2 x N2). This is a sparse matrix, but it has always been considered as highly undesirable to work with a solution (even though it may be closed-form) where the matrix to be solved increases in order roughly as the fourth power of the number of mesh intervals. The new algebraic form for this closed-form solution involves a matrix of order (N x N), not (N2 x N2). The matrix is, however, a full matrix. What is done, essentially, is to solve simultaneously for all the flux values along the vertical centerline of the two-dimensional problem, and then use a reflective boundary condition across the core centerline, and then the difference equation itself (in vector form) as a set of flux-vector generating equations to generate the entire flux field, line by line. In solving for the first flux vector (at the x = o, or z = o, core centerline), the right-hand side of the matrix problem incorporates all of the source values in the entire problem space. The initial inversion of the full (N x N) matrix algebraically guarantees that the (M+1)th flux vector (on the problem space boundary) will go to zero. This matrix method for two-dimensional neutronic analysis was shown to work well in both cartesian and cylindrical coordinates.
Download or read book Iterative Solution of Elliptic Systems written by Eugene L. Wachspress and published by . This book was released on 1966 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Iterative Solution of Elliptic Difference Equations written by Bernard Friedman and published by . This book was released on 1957 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On Some Iterative Methods for Solving Elliptic Difference Equations written by Herbert Bishop Keller and published by . This book was released on 1957 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Transport Phenomena written by Estéban Saatdjian and published by John Wiley & Sons. This book was released on 2000-11-08 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable text, provides a much-needed overview of both the theoretical development, as well as appropriate numerical solutions, for all aspects of transport phenomena. It contains a basic introduction to many aspects of fluid mechanics, heat transfer and mass transfer, and the conservation equations for mass, energy and momentum are discussed with reference to engineering applications. Heat transfer by conduction, radiation, natural and forced convection is studied, as well as mass transfer and incompressible fluid mechanics. The second part of the book deals with numerical methods used to solve the problems encountered earlier. The basic concepts of finite difference and finite volume methods are presented. Other subjects usually covered in mathematical textbooks such as vector and tensor analysis, Laplace transforms, and Runge-Kutta methods are discussed in the Appendices. * Offers comprehensive coverage of both transport phenomena and numerical and analytical solutions to the problems. * Includes comprehensive coverage of numerical techniques. * Provides real-life problems and solutions, which are vital to the understanding and implementation of applications. This work will be welcomed not only by senior and graduate students in mechanical, aeronautical and chemical engineering, but also for engineers practising in these fields.
Download or read book Iterative Methods and Preconditioners for Systems of Linear Equations written by Gabriele Ciaramella and published by SIAM. This book was released on 2022-02-08 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative methods use successive approximations to obtain more accurate solutions. This book gives an introduction to iterative methods and preconditioning for solving discretized elliptic partial differential equations and optimal control problems governed by the Laplace equation, for which the use of matrix-free procedures is crucial. All methods are explained and analyzed starting from the historical ideas of the inventors, which are often quoted from their seminal works. Iterative Methods and Preconditioners for Systems of Linear Equations grew out of a set of lecture notes that were improved and enriched over time, resulting in a clear focus for the teaching methodology, which derives complete convergence estimates for all methods, illustrates and provides MATLAB codes for all methods, and studies and tests all preconditioners first as stationary iterative solvers. This textbook is appropriate for undergraduate and graduate students who want an overview or deeper understanding of iterative methods. Its focus on both analysis and numerical experiments allows the material to be taught with very little preparation, since all the arguments are self-contained, and makes it appropriate for self-study as well. It can be used in courses on iterative methods, Krylov methods and preconditioners, and numerical optimal control. Scientists and engineers interested in new topics and applications will also find the text useful.
Download or read book A Comparison of Iterative Methods for a Model Coupled System of Elliptic Equations written by and published by . This book was released on 1993 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many interesting areas of current industry work deal with non-linear coupled systems of partial differential equations. We examine iterative methods for the solution of a model two-dimensional coupled system based on a linearized form of the two carrier drift-diffusion equations from semiconductor modeling. Discretizing this model system yields a large non-symmetric indefinite sparse matrix. To solve the model system various point and block methods, including the hybrid iterative method Alternate Block Factorization (ABF), are applied. We also employ GMRES with various preconditioners, including block and point incomplete LU (ILU) factorizations. The performance of these methods is compared. It is seen that the preferred ordering of the grid variables and the choice of iterative method are dependent upon the magnitudes of the coupling parameters. For this model, ABF is the most robust of the non-accelerated iterative methods. Among the preconditioners employed with GMRES, the blocked ''by grid point'' version of both the ILU and MILU preconditioners are the most robust and the most time efficient over the wide range of parameter values tested. This information may aid in the choice of iterative methods and preconditioners for solving more complicated, yet analogous, coupled systems.
Download or read book Air Force Research Resum s written by and published by . This book was released on with total page 572 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Energy Research Abstracts written by and published by . This book was released on 1993 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Iterative Methods for Large Linear Systems written by David R. Kincaid and published by Academic Press. This book was released on 2014-05-10 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative Methods for Large Linear Systems contains a wide spectrum of research topics related to iterative methods, such as searching for optimum parameters, using hierarchical basis preconditioners, utilizing software as a research tool, and developing algorithms for vector and parallel computers. This book provides an overview of the use of iterative methods for solving sparse linear systems, identifying future research directions in the mainstream of modern scientific computing with an eye to contributions of the past, present, and future. Different iterative algorithms that include the successive overrelaxation (SOR) method, symmetric and unsymmetric SOR methods, local (ad-hoc) SOR scheme, and alternating direction implicit (ADI) method are also discussed. This text likewise covers the block iterative methods, asynchronous iterative procedures, multilevel methods, adaptive algorithms, and domain decomposition algorithms. This publication is a good source for mathematicians and computer scientists interested in iterative methods for large linear systems.
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1992 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Iterative Methods for Solving Elliptic type Differential Equations with Application to Two space dimension Multigroup Analysis written by Eugene L. Wachspress and published by . This book was released on 1955 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Solution of Elliptic and Parabolic Partial Differential Equations with CD ROM written by John A. Trangenstein and published by Cambridge University Press. This book was released on 2013-04-18 with total page 657 pages. Available in PDF, EPUB and Kindle. Book excerpt: For mathematicians and engineers interested in applying numerical methods to physical problems this book is ideal. Numerical ideas are connected to accompanying software, which is also available online. By seeing the complete description of the methods in both theory and implementation, students will more easily gain the knowledge needed to write their own application programs or develop new theory. The book contains careful development of the mathematical tools needed for analysis of the numerical methods, including elliptic regularity theory and approximation theory. Variational crimes, due to quadrature, coordinate mappings, domain approximation and boundary conditions, are analyzed. The claims are stated with full statement of the assumptions and conclusions, and use subscripted constants which can be traced back to the origination (particularly in the electronic version, which can be found on the accompanying CD-ROM).
Download or read book Iterative Methods for Solving Partial Difference Equations of Elliptic Types written by David M. Young and published by . This book was released on 1950 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Iterative Methods for Linear and Nonlinear Equations written by C. T. Kelley and published by SIAM. This book was released on 1995-01-01 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Iterative Methods for Non linear Partial Differential Equations written by J. M. L. Maubach and published by . This book was released on 1994 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Methods for Solving Linear Systems and Applications to Elliptic Difference Equations written by Clarence Edgar Lee and published by . This book was released on 1959 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative numerical methods for solving independent, simultaneous, inhomogeneous linear equations are surveyed. Application of the methods to elliptic difference equations as arise in neutron diffasion, heat conduction, and potential problems is discussed.
