## Discretization Methods for Stable Initial Value Problems

Stability of explicit time-discretizations for solving initial value problems. Numer. Math. 37, 61-91 (1981). ... JENSEN, P.S. : Stiffly stable methods for undamped second order equations of motion. SIAM J. Numer. Anal.

Author: E. Gekeler

Publisher: Springer

ISBN: 9783540387633

Category: Mathematics

Page: 201

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## Analysis of Discretization Methods for Ordinary Differential Equations

In this situation, the classical text book by P.

Author: Hans J. Stetter

Publisher: Springer Science & Business Media

ISBN: 9783642654718

Category: Mathematics

Page: 390

View: 580

Due to the fundamental role of differential equations in science and engineering it has long been a basic task of numerical analysts to generate numerical values of solutions to differential equations. Nearly all approaches to this task involve a "finitization" of the original differential equation problem, usually by a projection into a finite-dimensional space. By far the most popular of these finitization processes consists of a reduction to a difference equation problem for functions which take values only on a grid of argument points. Although some of these finite difference methods have been known for a long time, their wide applica bility and great efficiency came to light only with the spread of electronic computers. This in tum strongly stimulated research on the properties and practical use of finite-difference methods. While the theory or partial differential equations and their discrete analogues is a very hard subject, and progress is consequently slow, the initial value problem for a system of first order ordinary differential equations lends itself so naturally to discretization that hundreds of numerical analysts have felt inspired to invent an ever-increasing number of finite-difference methods for its solution. For about 15 years, there has hardly been an issue of a numerical journal without new results of this kind; but clearly the vast majority of these methods have just been variations of a few basic themes. In this situation, the classical text book by P.
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## The Finite Element Method for Initial Value Problems

stability for select choices of discretization parameters in space and time and the dimensionless parameters in the mathematical model, then the method of approximation is termed conditionally stable. The third category of methods are ...

Author: Karan S. Surana

Publisher: CRC Press

ISBN: 9781351269995

Category: Science

Page: 630

View: 218

Unlike most finite element books that cover time dependent processes (IVPs) in a cursory manner, The Finite Element Method for Initial Value Problems: Mathematics and Computations focuses on the mathematical details as well as applications of space-time coupled and space-time decoupled finite element methods for IVPs. Space-time operator classification, space-time methods of approximation, and space-time calculus of variations are used to establish unconditional stability of space-time methods during the evolution. Space-time decoupled methods are also presented with the same rigor. Stability of space-time decoupled methods, time integration of ODEs including the finite element method in time are presented in detail with applications. Modal basis, normal mode synthesis techniques, error estimation, and a posteriori error computations for space-time coupled as well as space-time decoupled methods are presented. This book is aimed at a second-semester graduate level course in FEM.
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## Numerical Solutions of the N Body Problem

... discrete methods is a stability of them. From (1.2.2) it follows that a total discretization error is a difference between the solutions of equations Fn£n=0 and Fn: "n(F) Anz. If we assume that the initial value problem has a unique ...

Author: A. Marciniak

Publisher: Springer Science & Business Media

ISBN: 9789400954120

Category: Computers

Page: 242

View: 888

Approach your problem from the right It isn't that they can't see end and begin with the answers. the solution. Then one day, perhaps you will find It is that they can't see the the final question. problem. G.K. Chesterton. The Scandal The Hermit Clad in Crane Feathers in of Father Brown The Point of R. van Gulik's The Chinese Maze Murders. a Pin. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new brancheq. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics, algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisci fI plines as "experimental mathematics", "CFD , "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes.
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## A First Course in the Numerical Analysis of Differential Equations

Gekeler, E. (1984), Discretization Methods for Stable Initial Value Problems, Springer-Verlag, Berlin. Gottlieb, D. and Orszag, S.A. (1977), Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, ...

Author: Arieh Iserles

Publisher: Cambridge University Press

ISBN: 9781139473767

Category: Mathematics

Page:

View: 583

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.
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## Numerical Methods for Initial Value Problems in Ordinary Differential Equations

... attention to these special problem areas, which have attracted a lot of research attention within the last decade. 2.4. Error Propagation, Stability, and Convergence of Discretization Methods The integral curve (local solution) y(x ...

Author: Simeon Ola Fatunla

ISBN: 9781483269269

Category: Mathematics

Page: 308

View: 378

Numerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular, and discontinuous initial value problems, characterized by large Lipschitz constants. The book reviews the difference operators, the theory of interpolation, first integral mean value theorem, and numerical integration algorithms. The text explains the theory of one-step methods, the Euler scheme, the inverse Euler scheme, and also Richardson's extrapolation. The book discusses the general theory of Runge-Kutta processes, including the error estimation, and stepsize selection of the R-K process. The text evaluates the different linear multistep methods such as the explicit linear multistep methods (Adams-Bashforth, 1883), the implicit linear multistep methods (Adams-Moulton scheme, 1926), and the general theory of linear multistep methods. The book also reviews the existing stiff codes based on the implicit/semi-implicit, singly/diagonally implicit Runge-Kutta schemes, the backward differentiation formulas, the second derivative formulas, as well as the related extrapolation processes. The text is intended for undergraduates in mathematics, computer science, or engineering courses, andfor postgraduate students or researchers in related disciplines.
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## Solving Ordinary Differential Equations II

E. Gekeler ( 1984 ) : Discretization Methods for Stable Initial Value Problems . Lecture Notes in Math . , No. 1044 , Springer Verlag . ( V.7 ] Y. Genin ( 1974 ) : An algebraic approach to A - stable linear multistep - multiderivative ...

Author: Ernst Hairer

Publisher: Springer Science & Business Media

ISBN: 3540604529

Category: Mathematics

Page: 614

View: 844

"Whatever regrets may be, we have done our best." (Sir Ernest Shack 0 leton, turning back on 9 January 1909 at 88 23' South.) Brahms struggled for 20 years to write his first symphony. Compared to this, the 10 years we have been working on these two volumes may even appear short. This second volume treats stiff differential equations and differential algebraic equations. It contains three chapters: Chapter IV on one-step (Runge-Kutta) meth ods for stiff problems, Chapter V on multistep methods for stiff problems, and Chapter VI on singular perturbation and differential-algebraic equations. Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretical nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered con secutively in each section and indicate, in addition, the section number. In cross references to other chapters the (latin) chapter number is put first. References to the bibliography are again by "author" plus "year" in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete.
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## Theory of Difference Equations Numerical Methods and Applications by V Lakshmikantham and D Trigiante

[55] Gear, G. W. and Tu K. W., The effect of variable mesh size on the stability of multistep methods, SIAM JNA, 1 (1974), pp. 1025-1043. [56] Gekerel, E., Discretization methods for stable initial value problems, Lecture Notes in Math.

Author: Lakshmikantham

Publisher: Elsevier

ISBN: 0080958699

Category: Mathematics

Page: 322

View: 857

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering
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## Applications of Computer Technology to Dynamical Astronomy

(C) Gear, C.W., 1971, Numerical Initial Value Problems in Ordinary Differentail Equations, Prentice=hall, Englewood Cliffs. (A) Gekeler, E., 1984, Discretization Methods for Stable Initial Value Problems, Springer-Verlag.

Author: P. Kenneth Seidelmann

Publisher: Springer Science & Business Media

ISBN: 9789400909854

Category: Science

Page: 341

View: 363

Proceedings of the 109th Colloquium of the International Astronomical Union, held in Gaithersburg, Maryland, 27-29 July, 1988
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## Mathematical Methods for Mechanics

Gekeler, E.W.: Discretization Methods for Stable Initial Value Problems. Lecture Notes Math. Bd. 1044, Springer, Berlin Heidelberg New York (1984) Gekeler86. Gekeler, E.W., Widmann, R.: On the order conditions of Runge-Kutta methods ...

Author: Eckart W. Gekeler

Publisher: Springer Science & Business Media

ISBN: 9783540692799

Category: Technology & Engineering

Page: 624

View: 956

Mathematics is undoubtedly the key to state-of-the-art high technology. It is aninternationaltechnicallanguageandprovestobeaneternallyyoungscience to those who have learned its ways. Long an indispensable part of research thanks to modeling and simulation, mathematics is enjoying particular vit- ity now more than ever. Nevertheless, this stormy development is resulting in increasingly high requirements for students in technical disciplines, while general interest in mathematics continues to wane at the same time. This book and its appendices on the Internet seek to deal with this issue, helping students master the di?cult transition from the receptive to the productive phase of their education. The author has repeatedly held a three-semester introductory course - titled Higher Mathematics at the University of Stuttgart and used a series of “handouts” to show further aspects, make the course contents more motiv- ing, and connect with the mechanics lectures taking place at the same time. One part of the book has more or less evolved from this on its own. True to the original objective, this part treats a variety of separate topics of varying degrees of di?culty; nevertheless, all these topics are oriented to mechanics. Anotherpartofthisbookseekstoo?eraselectionofunderstandablereal- ticmodelsthatcanbeimplementeddirectlyfromthemultitudeofmathema- calresources.TheauthordoesnotattempttohidehispreferenceofNumerical Mathematics and thus places importance on careful theoretical preparation.
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