Numerical Methods for Evolutionary Differential Equations

Author: Uri M. Ascher

Publisher: SIAM

ISBN: 0898718910

Category: Evolution equations

Page: 395

View: 5213

Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. Audience: suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method.

Numerical Methods for Conservation Laws

From Analysis to Algorithms

Author: Jan S. Hesthaven

Publisher: SIAM

ISBN: 1611975107

Category: Science

Page: 570

View: 7384

Conservation laws are the mathematical expression of the principles of conservation and provide effective and accurate predictive models of our physical world. Although intense research activity during the last decades has led to substantial advances in the development of powerful computational methods for conservation laws, their solution remains a challenge and many questions are left open; thus it is an active and fruitful area of research. Numerical Methods for Conservation Laws: From Analysis to Algorithms offers the first comprehensive introduction to modern computational methods and their analysis for hyperbolic conservation laws, building on intense research activities for more than four decades of development; discusses classic results on monotone and finite difference/finite volume schemes, but emphasizes the successful development of high-order accurate methods for hyperbolic conservation laws; addresses modern concepts of TVD and entropy stability, strongly stable Runge-Kutta schemes, and limiter-based methods before discussing essentially nonoscillatory schemes, discontinuous Galerkin methods, and spectral methods; explores algorithmic aspects of these methods, emphasizing one- and two-dimensional problems and the development and analysis of an extensive range of methods; includes MATLAB software with which all main methods and computational results in the book can be reproduced; and demonstrates the performance of many methods on a set of benchmark problems to allow direct comparisons. Code and other supplemental material will be available online at publication.

Numerical Solution of Partial Differential Equations in Science and Engineering

Author: Leon Lapidus,George F. Pinder

Publisher: Wiley-Interscience


Category: Mathematics

Page: 677

View: 9678

"This book was written to provide a text for graduate and undergraduate students who took our courses in numerical methods. It incorporates the essential elements of all the numerical methods currently used extensively in the solution of partial differential equations encountered regularly in science and engineering. Because our courses were typically populated by students from varied backgrounds and with diverse interests, we attempted to eliminate jargon or nomenclature that would render the work unintelligible to any student. Moreover, in response to student needs, we incorporated not only classical (and not so classical) finite-difference methods but also finite-element, collocation, and boundary-element procedures. After an introduction to the various numerical schemes, each equation type--parabolic, elliptic, and hyperbolic--is allocated a separate chapter. Within each of these chapters the material is presented by numerical method. Thus one can read the book either by equation-type or numerical approach."--Preface, page [v].

Formulation and Numerical Solution of Quantum Control Problems

Author: Alfio Borzi,Gabriele Ciaramella,Martin Sprengel

Publisher: SIAM

ISBN: 1611974844

Category: Technology & Engineering

Page: 390

View: 1994

This book provides an introduction to representative nonrelativistic quantum control problems and their theoretical analysis and solution via modern computational techniques. The quantum theory framework is based on the SchrÓdinger picture, and the optimization theory, which focuses on functional spaces, is based on the Lagrange formalism. The computational techniques represent recent developments that have resulted from combining modern numerical techniques for quantum evolutionary equations with sophisticated optimization schemes. Both finite and infinite-dimensional models are discussed, including the three-level Lambda system arising in quantum optics, multispin systems in NMR, a charged particle in a well potential, Bose?Einstein condensates, multiparticle spin systems, and multiparticle models in the time-dependent density functional framework. This self-contained book covers the formulation, analysis, and numerical solution of quantum control problems and bridges scientific computing, optimal control and exact controllability, optimization with differential models, and the sciences and engineering that require quantum control methods.

Numerical Methods for Ordinary Differential Equations

Author: J. C. Butcher

Publisher: John Wiley & Sons

ISBN: 1119121507

Category: Mathematics

Page: 538

View: 6746

A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world’s leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numerical schemes to a source of new and practical algorithms in their own right. As the founder of general linear method research, John Butcher has been a leading contributor to its development; his special role is reflected in the text. The book is written in the lucid style characteristic of the author, and combines enlightening explanations with rigorous and precise analysis. In addition to these anticipated features, the book breaks new ground by including the latest results on the highly efficient G-symplectic methods which compete strongly with the well-known symplectic Runge-Kutta methods for long-term integration of conservative mechanical systems. Key features: ?? Presents a comprehensive and detailed study of the subject ?? Covers both practical and theoretical aspects ?? Includes widely accessible topics along with sophisticated and advanced details ?? Offers a balance between traditional aspects and modern developments This third edition of Numerical Methods for Ordinary Differential Equations will serve as a key text for senior undergraduate and graduate courses in numerical analysis, and is an essential resource for research workers in applied mathematics, physics and engineering.

Mathematical analysis and numerical methods for science and technology

Author: Robert Dautray,Jacques-Louis Lions

Publisher: Springer

ISBN: 9783540502067

Category: Mathematics

Page: 485

View: 4306

These six volumes - the result of a ten year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen " "Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to caluclate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every fact of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. "Volumes 5 and 6" cover problems of Transport and Evolution.

Scientific Computing with Ordinary Differential Equations

Author: Peter Deuflhard,Folkmar Bornemann

Publisher: Springer Science & Business Media

ISBN: 9780387954622

Category: Mathematics

Page: 486

View: 3118

Well-known authors; Includes topics and results that have previously not been covered in a book; Uses many interesting examples from science and engineering; Contains numerous homework exercises; Scientific computing is a hot and topical area

Spectral and High Order Methods for Partial Differential Equations

Selected papers from the ICOSAHOM '09 conference, June 22-26, Trondheim, Norway

Author: Jan S. Hesthaven,Einar M. Rønquist

Publisher: Springer Science & Business Media

ISBN: 9783642153372

Category: Mathematics

Page: 510

View: 4676

The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2009), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of the papers will provide the reader with a snapshot of state-of-the-art and help initiate new research directions through the extensive bibliography.

Numerical Methods for Nonlinear Partial Differential Equations

Author: Sören Bartels

Publisher: Springer

ISBN: 3319137972

Category: Mathematics

Page: 393

View: 2937

The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.

Partielle Differentialgleichungen und numerische Methoden

Author: Stig Larsson,Vidar Thomee

Publisher: Springer-Verlag

ISBN: 3540274227

Category: Mathematics

Page: 272

View: 5127

Das Buch ist für Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Für jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthält der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem für gewöhnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhängigen Problemen ein Kapitel zum Anfangswertproblem für gewöhnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen über lineare Funktionalanalysis und Sobolev-Räume wird im Anhang im Überblick besprochen.

Decomposition Methods for Differential Equations

Theory and Applications

Author: Juergen Geiser

Publisher: CRC Press

ISBN: 9781439810972

Category: Mathematics

Page: 304

View: 7884

Decomposition Methods for Differential Equations: Theory and Applications describes the analysis of numerical methods for evolution equations based on temporal and spatial decomposition methods. It covers real-life problems, the underlying decomposition and discretization, the stability and consistency analysis of the decomposition methods, and numerical results. The book focuses on the modeling of selected multi-physics problems, before introducing decomposition analysis. It presents time and space discretization, temporal decomposition, and the combination of time and spatial decomposition methods for parabolic and hyperbolic equations. The author then applies these methods to numerical problems, including test examples and real-world problems in physical and engineering applications. For the computational results, he uses various software tools, such as MATLAB®, R3T, WIAS-HiTNIHS, and OPERA-SPLITT. Exploring iterative operator-splitting methods, this book shows how to use higher-order discretization methods to solve differential equations. It discusses decomposition methods and their effectiveness, combination possibility with discretization methods, multi-scaling possibilities, and stability to initial and boundary values problems.

Enabling Technologies for Computational Science

Frameworks, Middleware and Environments

Author: Elias N. Houstis,John R. Rice,Efstratios Gallopoulos,Randall Bramley

Publisher: Springer Science & Business Media

ISBN: 9780792378099

Category: Computers

Page: 402

View: 5225

Enabling Technologies for Computational Science assesses future application computing needs, identifies research directions in problem-solving environments (PSEs), addresses multi-disciplinary environments operating on the Web, proposes methodologies and software architectures for building adaptive and human-centered PSEs, and describes the role of symbolic computing in scientific and engineering PSEs. The book also includes an extensive bibliography of over 400 references. Enabling Technologies for Computational Science illustrates the extremely broad and interdisciplinary nature of the creation and application of PSEs. Authors represent academia, government laboratories and industry, and come from eight distinct disciplines (chemical engineering, computer science, ecology, electrical engineering, mathematics, mechanical engineering, psychology and wood sciences). This breadth and diversity extends into the computer science aspects of PSEs. These papers deal with topics such as artificial intelligence, computer-human interaction, control, data mining, graphics, language design and implementation, networking, numerical analysis, performance evaluation, and symbolic computing. Enabling Technologies for Computational Science provides an assessment of the state of the art and a road map to the future in the area of problem-solving environments for scientific computing. This book is suitable as a reference for scientists from a variety of disciplines interested in using PSEs for their research.

Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016

Author: Zhongyi Huang,Martin Stynes,Zhimin Zhang

Publisher: Springer

ISBN: 3319672029

Category: Mathematics

Page: 211

View: 2973

This volume collects papers associated with lectures that were presented at the BAIL 2016 conference, which was held from 14 to 19 August 2016 at Beijing Computational Science Research Center and Tsinghua University in Beijing, China. It showcases the variety and quality of current research into numerical and asymptotic methods for theoretical and practical problems whose solutions involve layer phenomena. The BAIL (Boundary And Interior Layers) conferences, held usually in even-numbered years, bring together mathematicians and engineers/physicists whose research involves layer phenomena, with the aim of promoting interaction between these often-separate disciplines. These layers appear as solutions of singularly perturbed differential equations of various types, and are common in physical problems, most notably in fluid dynamics. This book is of interest for current researchers from mathematics, engineering and physics whose work involves the accurate app roximation of solutions of singularly perturbed differential equations; that is, problems whose solutions exhibit boundary and/or interior layers.

Numerical Methods for Viscosity Solutions and Applications

Author: Maurizio Falcone,Charalampos Makridakis

Publisher: World Scientific

ISBN: 9789812799807

Category: Mathematics

Page: 248

View: 8583

The volume contains twelve papers dealing with the approximation of first and second order problems which arise in many fields of application including optimal control, image processing, geometrical optics and front propagation. Some contributions deal with new algorithms and technical issues related to their implementation. Other contributions are more theoretical, dealing with the convergence of approximation schemes. Many test problems have been examined to evaluate the performances of the algorithms. The volume can attract readers involved in the numerical approximation of differential models in the above-mentioned fields of applications, engineers, graduate students as well as researchers in numerical analysis. Contents: Geometrical Optics and Viscosity Solutions (A-P Blanc et al.); Computation of Vorticity Evolution for a Cylindrical Type-II Superconductor Subject to Parallel and Transverse Applied Magnetic Fields (A Briggs et al.); A Characterization of the Value Function for a Class of Degenerate Control Problems (F Camilli); Some Microstructures in Three Dimensions (M Chipot & V L(r)cuyer); Convergence of Numerical Schemes for the Approximation of Level Set Solutions to Mean Curvature Flow (K Deckelnick & G Dziuk); Optimal Discretization Steps in Semi-Lagrangian Approximation of First Order PDEs (M Falcone et al.); Convergence Past Singularities to the Forced Mean Curvature Flow for a Modified Reaction-Diffusion Approach (F Fierro); The Viscosity/Duality Solutions Approach to Geometric Optics for the Helmholtz Equation (L Gosse & F James); Adaptive Grid Generation for Evolutive Hamilton-Jacobi-Bellman Equations (L Grne); Solution and Application of Anisotropic Curvature Driven Evolution of Curves (and Surfaces) (K Mikula); An Adaptive Scheme on Unstructured Grids for the Shape-From-Shading Problem (M Sagona & A Seghini); On a Posteriori Error Estimation for Constant Obstacle Problems (A Veeser). Readership: Graduate students, researchers, academics and lecturers in numerical & computational mathematics, analysis & differential equations and mathematical modeling.

Integral Methods in Science and Engineering

Computational and Analytic Aspects

Author: Christian Constanda,Paul J. Harris

Publisher: Springer Science & Business Media

ISBN: 0817682384

Category: Mathematics

Page: 422

View: 6727

An enormous array of problems encountered by scientists and engineers are based on the design of mathematical models using many different types of ordinary differential, partial differential, integral, and integro-differential equations. Accordingly, the solutions of these equations are of great interest to practitioners and to science in general. Presenting a wealth of cutting-edge research by a diverse group of experts in the field, Integral Methods in Science and Engineering: Computational and Analytic Aspects gives a vivid picture of both the development of theoretical integral techniques and their use in specific science and engineering problems. This book will be valuable for researchers in applied mathematics, physics, and mechanical and electrical engineering. It will likewise be a useful study guide for graduate students in these disciplines, and for various other professionals who use integration as an essential technique in their work.

Data-Driven Modeling & Scientific Computation

Methods for Complex Systems & Big Data

Author: J. Nathan Kutz

Publisher: Oxford University Press

ISBN: 0199660336

Category: Computers

Page: 638

View: 2641

Combining scientific computing methods and algorithms with modern data analysis techniques, including basic applications of compressive sensing and machine learning, this book develops techniques that allow for the integration of the dynamics of complex systems and big data. MATLAB is used throughout for mathematical solution strategies.

Introduction to Numerical Programming

A Practical Guide for Scientists and Engineers Using Python and C/C++

Author: Titus A. Beu

Publisher: CRC Press

ISBN: 1466569670

Category: Mathematics

Page: 674

View: 1763

Makes Numerical Programming More Accessible to a Wider Audience Bearing in mind the evolution of modern programming, most specifically emergent programming languages that reflect modern practice, Numerical Programming: A Practical Guide for Scientists and Engineers Using Python and C/C++ utilizes the author’s many years of practical research and teaching experience to offer a systematic approach to relevant programming concepts. Adopting a practical, broad appeal, this user-friendly book offers guidance to anyone interested in using numerical programming to solve science and engineering problems. Emphasizing methods generally used in physics and engineering—from elementary methods to complex algorithms—it gradually incorporates algorithmic elements with increasing complexity. Develop a Combination of Theoretical Knowledge, Efficient Analysis Skills, and Code Design Know-How The book encourages algorithmic thinking, which is essential to numerical analysis. Establishing the fundamental numerical methods, application numerical behavior and graphical output needed to foster algorithmic reasoning, coding dexterity, and a scientific programming style, it enables readers to successfully navigate relevant algorithms, understand coding design, and develop efficient programming skills. The book incorporates real code, and includes examples and problem sets to assist in hands-on learning. Begins with an overview on approximate numbers and programming in Python and C/C++, followed by discussion of basic sorting and indexing methods, as well as portable graphic functionality Contains methods for function evaluation, solving algebraic and transcendental equations, systems of linear algebraic equations, ordinary differential equations, and eigenvalue problems Addresses approximation of tabulated functions, regression, integration of one- and multi-dimensional functions by classical and Gaussian quadratures, Monte Carlo integration techniques, generation of random variables, discretization methods for ordinary and partial differential equations, and stability analysis This text introduces platform-independent numerical programming using Python and C/C++, and appeals to advanced undergraduate and graduate students in natural sciences and engineering, researchers involved in scientific computing, and engineers carrying out applicative calculations.

Implementing Spectral Methods for Partial Differential Equations

Algorithms for Scientists and Engineers

Author: David A. Kopriva

Publisher: Springer Science & Business Media

ISBN: 9048122619

Category: Mathematics

Page: 397

View: 6903

This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.

Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

Author: Graham Griffiths,William E. Schiesser

Publisher: Academic Press

ISBN: 9780123846532

Category: Mathematics

Page: 461

View: 2270

Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods. This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors’ intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs. The Matlab and Maple software will be available for download from this website shortly. Includes a spectrum of applications in science, engineering, applied mathematics Presents a combination of numerical and analytical methods Provides transportable computer codes in Matlab and Maple