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The Importance Of Boundary Conditions In The Numerical Treatment Of Hyperbolic Equations
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Book Synopsis The Importance of Boundary Conditions in the Numerical Treatment of Hyperbolic Equations by : Gino Moretti
Download or read book The Importance of Boundary Conditions in the Numerical Treatment of Hyperbolic Equations written by Gino Moretti and published by . This book was released on 1968 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many of the existing computations of initial-and-boundary value problems in fluid mechanics suffer from unrealistic treatment of boundary points. Three categories of boundaries are briefly discussed: rigid walls, arbitrary boundaries of a computational region in a subsonic flow, and shock waves. An attempt is made to show in what sense the numerical treatment of such boundaries may be physically wrong and what can be done instead. Examples from the blunt body problem, the transonic flow in a nozzle, the incompressible inviscid flow past a circle, and the quasi-one-dimensional flow in a Laval nozzle, are shown. (Author).
Book Synopsis Numerical Methods in Fluid Dynamics by : Gary A. Sod
Download or read book Numerical Methods in Fluid Dynamics written by Gary A. Sod and published by Cambridge University Press. This book was released on 1985-10-31 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is an introduction to numerical methods for partial differential equations with particular reference to those that are of importance in fluid dynamics. The author gives a thorough and rigorous treatment of the techniques, beginning with the classical methods and leading to a discussion of modern developments. For easier reading and use, many of the purely technical results and theorems are given separately from the main body of the text. The presentation is intended for graduate students in applied mathematics, engineering and physical sciences who have a basic knowledge of partial differential equations.
Book Synopsis Hyperbolic Boundary Value Problems by : Reiko Sakamoto
Download or read book Hyperbolic Boundary Value Problems written by Reiko Sakamoto and published by CUP Archive. This book was released on 1982-07-08 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Boundary value problems are of central importance and interest not only to mathematicians but also to physicists and engineers who need to solve differential equations which govern the behaviour of physical systems. In this book, Professor Sakamoto introduces the general theory of the existence and uniqueness of solutions to the wave equation. The reader is assumed to have some familiarity with Lebesgue integration and complex function theory but other than that the book is essentially self-contained. It is therefore suited to senior undergraduates and graduates in mathematics and the mathematical sciences but can be read with profit by professionals in those subjects.
Book Synopsis Handbook of Numerical Methods for Hyperbolic Problems by : Remi Abgrall
Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2017-01-16 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage
Book Synopsis Mathematical Aspects of Numerical Solution of Hyperbolic Systems by : A.G. Kulikovskii
Download or read book Mathematical Aspects of Numerical Solution of Hyperbolic Systems written by A.G. Kulikovskii and published by CRC Press. This book was released on 2000-12-21 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena. The authors also address a number of original "nonclassical" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice. This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.
Book Synopsis Solving Hyperbolic Equations with Finite Volume Methods by : M. Elena Vázquez-Cendón
Download or read book Solving Hyperbolic Equations with Finite Volume Methods written by M. Elena Vázquez-Cendón and published by Springer. This book was released on 2015-04-16 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite volume methods are used in numerous applications and by a broad multidisciplinary scientific community. The book communicates this important tool to students, researchers in training and academics involved in the training of students in different science and technology fields. The selection of content is based on the author’s experience giving PhD and master courses in different universities. In the book the introduction of new concepts and numerical methods go together with simple exercises, examples and applications that contribute to reinforce them. In addition, some of them involve the execution of MATLAB codes. The author promotes an understanding of common terminology with a balance between mathematical rigor and physical intuition that characterizes the origin of the methods. This book aims to be a first contact with finite volume methods. Once readers have studied it, they will be able to follow more specific bibliographical references and use commercial programs or open source software within the framework of Computational Fluid Dynamics (CFD).
Book Synopsis Boundary Conditions in the Numerical Integration of Hyperbolic Equations by : Timothy May
Download or read book Boundary Conditions in the Numerical Integration of Hyperbolic Equations written by Timothy May and published by . This book was released on 1978 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Treatment of Lateral Boundary Conditions in Limited-area Models by : Ralph Shapiro
Download or read book The Treatment of Lateral Boundary Conditions in Limited-area Models written by Ralph Shapiro and published by . This book was released on 1977 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical models used to simulate or predict atmospheric behavior are limited in scope because of insufficient knowledge of the initial conditions and forcing functions. But these limitations are generally dwarfed by the limited capacity and power of even the largest and most powerful computers. Thus, where a solution is required with high resolution for some limited area of interest, it is often impractical to solve the model throughout the entire domain with the same uniformly high resolution. A pragmatic approach is proposed here which overspecifies the artificial lateral boundaries of a fine-mesh, limited-area, nested region and depends upon a high-order phase and amplitude restoring interpolation along with a high-order, but weak, smoothing operator to control noise and ensure versimilitude. The dynamic tests are carried out in the framework of a multilevel primitive equation model and show unequivocally the importance of correct phase information for the lateral boundaries of the fine-mesh region. The results obtained indicate that the simple procedure proposed here should yield satisfactory results in operational fine-mesh, limited-area models.
Book Synopsis Boundary Value Problems for Hyperbolic Partial Differential Equations with Constant Coefficients by : J. F. Heyda
Download or read book Boundary Value Problems for Hyperbolic Partial Differential Equations with Constant Coefficients written by J. F. Heyda and published by . This book was released on 1959 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Some Problems On Nonlinear Hyperbolic Equations And Applications by : Tatsien Li
Download or read book Some Problems On Nonlinear Hyperbolic Equations And Applications written by Tatsien Li and published by World Scientific. This book was released on 2010-09-21 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is composed of two parts: Mathematical and Numerical Analysis for Strongly Nonlinear Plasma Models and Exact Controllability and Observability for Quasilinear Hyperbolic Systems and Applications. It presents recent progress and results obtained in the domains related to both subjects without attaching much importance to the details of proofs but rather to difficulties encountered, to open problems and possible ways to be exploited. It will be very useful for promoting further study on some important problems in the future.
Book Synopsis Well-posedness of Linear Hyperbolic Problems by : Aleksandr Mikhaĭlovich Blokhin
Download or read book Well-posedness of Linear Hyperbolic Problems written by Aleksandr Mikhaĭlovich Blokhin and published by Nova Publishers. This book was released on 2006 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book will be useful for students and specialists of partial differential equations and the mathematical sciences because it clarifies crucial points of Kreiss' symmetrizer technique. The Kreiss technique was developed by H.O. Kreiss for initial boundary value problems for linear hyperbolic systems. This technique is important because it involves equations that are used in many of the applied sciences. The research presented in this book takes unique approaches to exploring the Kreiss technique that will add insight and new perspectives to linear hyperbolic problems"--Publ. web site.
Book Synopsis High-order finite difference approximations for hyperbolic problems by : Hannes Frenander
Download or read book High-order finite difference approximations for hyperbolic problems written by Hannes Frenander and published by Linköping University Electronic Press. This book was released on 2017-01-24 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.
Book Synopsis Time-Dependent Problems and Difference Methods by : Bertil Gustafsson
Download or read book Time-Dependent Problems and Difference Methods written by Bertil Gustafsson and published by John Wiley & Sons. This book was released on 2013-08-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the First Edition ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations." —SIAM Review Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.
Book Synopsis Numerical Methods for Hyperbolic Equations by : Elena Vázquez-Cendón
Download or read book Numerical Methods for Hyperbolic Equations written by Elena Vázquez-Cendón and published by CRC Press. This book was released on 2012-11-05 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods for Hyperbolic Equations is a collection of 49 articles presented at the International Conference on Numerical Methods for Hyperbolic Equations: Theory and Applications (Santiago de Compostela, Spain, 4-8 July 2011). The conference was organized to honour Professor Eleuterio Toro in the month of his 65th birthday. The topics covered include: • Recent advances in the numerical computation of environmental conservation laws with source terms • Multiphase flow and porous media • Numerical methods in astrophysics • Seismology and geophysics modelling • High order methods for hyperbolic conservation laws • Numerical methods for reactive flows • Finite volume and discontinous Galerkin schemes for stiff source term problems • Methods and models for biomedical problems • Numerical methods for reactive flows The research interest of Eleuterio Toro, born in Chile on 16th July 1946, is reflected in Numerical Methods for Hyperbolic Equations, and focuses on: numerical methods for partial differential equations, with particular emphasis on methods for hyperbolic equations; design and application of new algorithms; hyperbolic partial differential equations as mathematical models of various types of processes; mathematical modelling and simulation of physico/chemical processes that include wave propagation phenomena; modelling of multiphase flows; application of models and methods to real problems. Eleuterio Toro received several honours and distinctions, including the honorary title OBE from Queen Elizabeth II (Buckingham Palace, London 2000); Distinguished Citizen of the City of Carahue (Chile, 2001); Life Fellow, Claire Hall, University of Cambridge (UK, 2003); Fellow of the Indian Society for Shock Wave Research (Bangalore, 2005); Doctor Honoris Causa (Universidad de Santiago de Chile, 2008); William Penney Fellow, University of Cambridge (UK, 2010); Doctor Honoris Causa (Universidad de la Frontera, Chile, 2012). Professor Toro is author of two books, editor of two books and author of more than 260 research works. In the last ten years he has been invited and keynote speaker in more than 100 scientific events. Professor Toro has held many visiting appointments round the world, which include several European countries, Japan, China and USA.
Book Synopsis Computational Gasdynamics by : Culbert B. Laney
Download or read book Computational Gasdynamics written by Culbert B. Laney and published by Cambridge University Press. This book was released on 1998-06-13 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical methods are indispensable tools in the analysis of complex fluid flows. This book focuses on computational techniques for high-speed gas flows, especially gas flows containing shocks and other steep gradients. The book decomposes complicated numerical methods into simple modular parts, showing how each part fits and how each method relates to or differs from others. The text begins with a review of gasdynamics and computational techniques. Next come basic principles of computational gasdynamics. The last two parts cover basic techniques and advanced techniques. Senior and graduate level students, especially in aerospace engineering, as well as researchers and practising engineers, will find a wealth of invaluable information on high-speed gas flows in this text.
Book Synopsis Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations by : A.K. Aziz
Download or read book Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations written by A.K. Aziz and published by Academic Press. This book was released on 2014-05-10 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations covers the proceedings of the 1974 Symposium by the same title, held at the University of Maryland, Baltimore Country Campus. This symposium aims to bring together a number of numerical analysis involved in research in both theoretical and practical aspects of this field. This text is organized into three parts encompassing 15 chapters. Part I reviews the initial and boundary value problems. Part II explores a large number of important results of both theoretical and practical nature of the field, including discussions of the smooth and local interpolant with small K-th derivative, the occurrence and solution of boundary value reaction systems, the posteriori error estimates, and boundary problem solvers for first order systems based on deferred corrections. Part III highlights the practical applications of the boundary value problems, specifically a high-order finite-difference method for the solution of two-point boundary-value problems on a uniform mesh. This book will prove useful to mathematicians, engineers, and physicists.
Book Synopsis Essential Partial Differential Equations by : David F. Griffiths
Download or read book Essential Partial Differential Equations written by David F. Griffiths and published by Springer. This book was released on 2015-09-24 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems. The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors. Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.
