Download or read book Oscillatory Solutions of the Peaceman Rachford Alternating Direction Implicit Method and a Comparison of Methods for the Solution of the Two Dimensional Heat Diffusion Equation written by R. F. Kropf and published by . This book was released on 1985 with total page 71 pages. Available in PDF, EPUB and Kindle. Book excerpt: The two dimensional heat diffusion equation with Dirichlet boundary conditions was solved using the fully explicit, fully implicit, Crank-Nicolson implicit, and Peaceman-Rachford alternating direction implicit (ADI) methods. Comparisons of accuracy and time requirements were made. The possibility that the ADI method has stable oscillatory solutions with large time steps was investigated. Results of computations revealed that the ADI has stable oscillations for large time steps, in some cases producing large enough errors to render the solution unusable. Time steps greater than twice the square of the mesh spacing divided by the thermal diffusivity must be used with care. For small time steps, the Crank-Nicolson and ADI methods were the most accurate, and the ADI was the fastest method. The fully implicit method was the most accurate at large time steps, but the ADI, with a smaller time step to reduce the oscillatory error, was still the fastest method to reach a solution with the desired degree of accuracy. Additional keywords: theses; numerical analysis; finite difference theory; conduction (heat transfer); partial differential equations; stability.(Author).
Download or read book Oscillatory Solutions of the Peaceman Rachford Alternating Direction Implicit Method and a Comparison of Methods for the Solution of the Two Dimensional Heat Diffusion Equation written by Roger F. Kropf (CAPT, USAF.) and published by . This book was released on 1985 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1985 with total page 740 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Sci tech Book Profiles written by and published by . This book was released on 1965 with total page 1208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes title page, table of contents, list of contributors, preface and all indexes of each book.
Download or read book Government reports annual index written by and published by . This book was released on 199? with total page 1228 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Comparison of Implicit Numerical Methods for Solving the Transient Spherical Diffusion Equation written by and published by . This book was released on 1977 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book International Aerospace Abstracts written by and published by . This book was released on 1985 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Oscillatory Behavior of Finite Difference Methods for the Solution of the Two Dimensional Transient Heat Diffusion Equation written by Joseph E. Cuthrell and published by . This book was released on 1986 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: The two dimensional transient heat (diffusion) equation with Dirichlet boundary conditions was solved using the Dufort-Frankel, Saul'ev, and, Exponential (Power-law) finite fidderence schemes. All methods were investigated for oscillatory behavior and comparisons of accuracy made. To predict the time step at which oscillatory behavior would occur, the coefficient, matrix, and probabilistic methods of stability analysis were utilized. At time steps greater than the square of the mesh divided by the thermal diffusivity, oscillatory solutions were apparent in both the Dufort-Frankel and Saul'ev schems. The exponential method, as predicted, did not oscillate for any size time step. Although the exponential scheme was the most accurate at large time steps, the solution still contained enough error to be unusable in many engineering applications. At small time steps, all methods were more accurate than the fully implicit formulation. The exponential method was found to be the slowest computationally. The Saul'ev scheme proved to be the fastest while still achieving the required degree of accuracy.
Download or read book Solution of Two dimensional Heat Problems Using the Alternating Direction Method written by Frederick James Leipold and published by . This book was released on 1967 with total page 89 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite difference approximations to the one-dimensional heat equation Ut = Uxx are used to introduce explicit and implicit difference equations. The convergence and stability of these equations is discussed and those concepts are used to show the restrictions imposed on explicit equations. The Implicit Alternating Direction (IAD) method is introduced as a means for solution of the two-dimensional heat equation. Although the IAD method in its basic form is applicable only to parabolic problems it is possible by slight modification to apply the method to elliptic problems. Two examples are used to illustrate the use of the IAD method for solution of parabolic and elliptic equations for a rectangular region. These examples include a work requirement comparision with other difference methods. A third example is given to show the applicability of the IAD method to non-rectangular regions, in this case a parabolic problem over a circular region. Results of these examples show that the IAD method is a convenient and accurate method when applied to both parabolic and elliptic partial differential equations and suggest applicability to a wide range of problems. (Author).
Download or read book Solution of Two dimensional Heat Problems Using the Alternating Direction Method written by and published by . This book was released on 1967 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite difference approximations to the one-dimensional heat equation Ut = Uxx are used to introduce explicit and implicit difference equations. The convergence and stability of these equations is discussed and those concepts are used to show the restrictions imposed on explicit equations. The Implicit Alternating Direction (IAD) method is introduced as a means for solution of the two-dimensional heat equation. Although the IAD method in its basic form is applicable only to parabolic problems it is possible by slight modification to apply the method to elliptic problems. Two examples are used to illustrate the use of the IAD method for solution of parabolic and elliptic equations for a rectangular region. These examples include a work requirement comparision with other difference methods. A third example is given to show the applicability of the IAD method to non-rectangular regions, in this case a parabolic problem over a circular region. Results of these examples show that the IAD method is a convenient and accurate method when applied to both parabolic and elliptic partial differential equations and suggest applicability to a wide range of problems. (Author).
Download or read book The Alternating Direction Implicit Numerical Solution of Time Dependent Two dimensional Two fluid Magnetohydrodynamic Equations written by Irvin Raymond Lindemuth and published by . This book was released on 1971 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Beyond Perturbation written by Shijun Liao and published by CRC Press. This book was released on 2003-10-27 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity. This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra's population model, Von Karman swirling viscous flow, and nonlinear progressive waves in deep water. Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open.
Download or read book Iterative Methods for Sparse Linear Systems written by Yousef Saad and published by SIAM. This book was released on 2003-04-01 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- General.
Download or read book Numerical Methods for Fluid Dynamics written by Dale R. Durran and published by Springer Science & Business Media. This book was released on 2010-09-14 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean
Download or read book The Mathematics of Diffusion written by John Crank and published by Oxford University Press. This book was released on 1979 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.
Download or read book Nonlinear Problems of Engineering written by William F. Ames and published by Academic Press. This book was released on 2014-05-12 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Problems of Engineering reviews certain nonlinear problems of engineering. This book provides a discussion of nonlinear problems that occur in four areas, namely, mathematical methods, fluid mechanics, mechanics of solids, and transport phenomena. Organized into 15 chapters, this book begins with an overview of some of the fundamental ideas of two mathematical theories, namely, invariant imbedding and dynamic programming. This text then explores nonlinear integral equations, which have long occupied a prominent place in mathematical analysis. Other chapters consider the phenomena associated with essentially divergent small-divisor series, such as may occur in the formal solution of differential equations that represent the oscillations of conservative dynamical systems. This book discusses as well the mechanics of idealized textiles consisting of inextensible filaments. The final chapter deals with the use of the Peaceman–Rachford alternating direction implicit method for solving the finite difference analogs of boundary value problems. This book is a valuable resource for engineers and mathematicians.
Download or read book Finite Difference Methods in Financial Engineering written by Daniel J. Duffy and published by John Wiley & Sons. This book was released on 2013-10-28 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method. In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature: Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options Early exercise features and approximation using front-fixing, penalty and variational methods Modelling stochastic volatility models using Splitting methods Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work Modelling jumps using Partial Integro Differential Equations (PIDE) Free and moving boundary value problems in QF Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.
