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Numerical Solution of Partial Differential Equations on Parallel Computers

Numerical Solution of Partial Differential Equations on Parallel Computers Author Are Magnus Bruaset
ISBN-10 9783540316190
Release 2006-03-05
Pages 482
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Since the dawn of computing, the quest for a better understanding of Nature has been a driving force for technological development. Groundbreaking achievements by great scientists have paved the way from the abacus to the supercomputing power of today. When trying to replicate Nature in the computer’s silicon test tube, there is need for precise and computable process descriptions. The scienti?c ?elds of Ma- ematics and Physics provide a powerful vehicle for such descriptions in terms of Partial Differential Equations (PDEs). Formulated as such equations, physical laws can become subject to computational and analytical studies. In the computational setting, the equations can be discreti ed for ef?cient solution on a computer, leading to valuable tools for simulation of natural and man-made processes. Numerical so- tion of PDE-based mathematical models has been an important research topic over centuries, and will remain so for centuries to come. In the context of computer-based simulations, the quality of the computed results is directly connected to the model’s complexity and the number of data points used for the computations. Therefore, computational scientists tend to ?ll even the largest and most powerful computers they can get access to, either by increasing the si e of the data sets, or by introducing new model terms that make the simulations more realistic, or a combination of both. Today, many important simulation problems can not be solved by one single computer, but calls for parallel computing.

Solving Partial Differential Equations on Parallel Computers

Solving Partial Differential Equations on Parallel Computers Author Jianping Zhu
ISBN-10 9789814522175
Release 1994-02-24
Pages 276
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This is an introductory book on supercomputer applications written by a researcher who is working on solving scientific and engineering application problems on parallel computers. The book is intended to quickly bring researchers and graduate students working on numerical solutions of partial differential equations with various applications into the area of parallel processing. The book starts from the basic concepts of parallel processing, like speedup, efficiency and different parallel architectures, then introduces the most frequently used algorithms for solving PDEs on parallel computers, with practical examples. Finally, it discusses more advanced topics, including different scalability metrics, parallel time stepping algorithms and new architectures and heterogeneous computing networks which have emerged in the last few years of high performance computing. Hundreds of references are also included in the book to direct interested readers to more detailed and in-depth discussions of specific topics. Contents:IntroductionParallel Algorithms for Solving PDEImplementationsApplicationsParallel Time Stepping AlgorithmsFuture Development Readership: Computer scientists, applied mathematicians, engineers and students. keywords:Parallel Computing;Partial Differential Equations;Numerical Algorithms for PDEs;Alternating Direction Implicit Algorithms;Parallel Computing and Applications

Parallel Solution of Partial Differential Equations

Parallel Solution of Partial Differential Equations Author Petter Bjorstad
ISBN-10 9781461211761
Release 2012-12-06
Pages 306
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This IMA Volume in Mathematics and its Applications PARALLEL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS is based on the proceedings of a workshop with the same title. The work shop was an integral part of the 1996-97IMA program on "MATHEMAT ICS IN HIGH-PERFORMANCE COMPUTING." I would like to thank Petter Bj0rstad of the Institutt for Informatikk, University of Bergen and Mitchell Luskin of the School of Mathematics, University of Minnesota for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Founda tion (NSF), Department of Energy (DOE), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE The numerical solution of partial differential equations has been of major importance to the development of many technologies and has been the target of much of the development of parallel computer hardware and software. Parallel computers offer the promise of greatly increased perfor mance and the routine calculation of previously intractable problems. The papers in this volume were presented at the IMA workshop on the Paral lel Solution of PDE held during June 9-13, 1997. The workshop brought together leading numerical analysts, computer scientists, and engineers to assess the state-of-the-art and to consider future directions.

Domain Decomposition

Domain Decomposition Author Barry Smith
ISBN-10 0521602866
Release 2004-03-25
Pages 240
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Presents an easy-to-read discussion of domain decomposition algorithms, their implementation and analysis. Ideal for graduate students about to embark on a career in computational science. It will also be a valuable resource for all those interested in parallel computing and numerical computational methods.

Parallel and Sequential Methods for Ordinary Differential Equations

Parallel and Sequential Methods for Ordinary Differential Equations Author Kevin Burrage
ISBN-10 9780198534327
Release 1995
Pages 446
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This book presents an up-to-date exposition of the current `state of the art' of numerical methods for solving ordinary differential equations in a parallel computing environment. Although the main focus is on problems of initial value type, consideration will also be given to boundary value problems and partial differential equations. Furthermore, because linear algebra is an important component of the solution of differential equations, a complete chapter is devoted to the parallel solution of linear systems of equations. In addition to presenting an overview of parallel computing in general, two chapters are devoted to a summary of existing sequential differential equation methods. The parallel techniques discussed include parallelism across the method, parallelism across the step, parallelism across the system, and dynamic iteration. The book concludes with a chapter on the behaviour of a parallel code based on waveform relaxation. This comprehensive book is unique in its content and provides a balance between theoretical and practical issues by providing general frameworks in which to study parallel methods.

Advanced Topics in Computational Partial Differential Equations

Advanced Topics in Computational Partial Differential Equations Author Hans Petter Langtangen
ISBN-10 9783642182372
Release 2012-09-22
Pages 663
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A gentle introduction to advanced topics such as parallel computing, multigrid methods, and special methods for systems of PDEs. The goal of all chapters is to ‘compute’ solutions to problems, hence algorithmic and software issues play a central role. All software examples use the Diffpack programming environment - some experience with Diffpack is required. There are also some chapters covering complete applications, i.e., the way from a model, expressed as systems of PDEs, through to discretization methods, algorithms, software design, verification, and computational examples. Suitable for readers with a background in basic finite element and finite difference methods for partial differential equations.

Group Explicit Methods for the Numerical Solution of Partial Differential Equations

Group Explicit Methods for the Numerical Solution of Partial Differential Equations Author David J. Evans
ISBN-10 9056990195
Release 1997-05-22
Pages 480
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A new class of methods, termed "group explicit methods," is introduced in this text. Their applications to solve parabolic, hyperbolic and elliptic equations are outlined, and the advantages for their implementation on parallel computers clearly portrayed. Also included are the introductory and fundamental concepts from which the new methods are derived, and on which they are dependent. With the increasing advent of parallel computing into all aspects of computational mathematics, there is no doubt that the new methods will be widely used.

Domain Decomposition Methods for Partial Differential Equations

Domain Decomposition Methods for Partial Differential Equations Author Alfio Quarteroni
ISBN-10 0198501781
Release 1999
Pages 360
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They comprise a relatively new field of study, but have already found applications in many branches of physics and engineering

Computational Partial Differential Equations

Computational Partial Differential Equations Author Hans Petter Langtangen
ISBN-10 9783662011706
Release 2013-04-17
Pages 685
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Targeted at students and researchers in computational sciences who need to develop computer codes for solving PDEs, the exposition here is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods, and basic finite difference methods from a computational point of view, with the main emphasis on developing flexible computer programs, using the numerical library Diffpack. Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. All the program examples, as well as Diffpack for use with this book, are available on the Internet. XXXXXXX NEUER TEXT This book is for researchers who need to develop computer code for solving PDEs. Numerical methods and the application of Diffpack are explained in detail. Diffpack is a modern C++ development environment that is widely used by industrial scientists and engineers working in areas such as oil exploration, groundwater modeling, and materials testing. All the program examples, as well as a test version of Diffpack, are available for free over the Internet.

Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations

Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations Author Tarek Mathew
ISBN-10 9783540772095
Release 2008-06-25
Pages 770
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Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative algorithms, and techniques for non-matching grid discretizations and heterogeneous approximations. This book serves as a matrix oriented introduction to domain decomposition methodology. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more.

A Tutorial on Elliptic PDE Solvers and Their Parallelization

A Tutorial on Elliptic PDE Solvers and Their Parallelization Author Craig C. Douglas
ISBN-10 0898718171
Release 2003-01-01
Pages 135
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This compact yet thorough tutorial is the perfect introduction to the basic concepts of solving partial differential equations (PDEs) using parallel numerical methods. In just eight short chapters, the authors provide readers with enough basic knowledge of PDEs, discretization methods, solution techniques, parallel computers, parallel programming, and the run-time behavior of parallel algorithms to allow them to understand, develop, and implement parallel PDE solvers. Examples throughout the book are intentionally kept simple so that the parallelization strategies are not dominated by technical details.

Finite Difference Methods for Ordinary and Partial Differential Equations

Finite Difference Methods for Ordinary and Partial Differential Equations Author Randall J. LeVeque
ISBN-10 0898717833
Release 2007-01-01
Pages 339
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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Solution of Partial Differential Equations on Vector and Parallel Computers

Solution of Partial Differential Equations on Vector and Parallel Computers Author James M. Ortega
ISBN-10 9780898710557
Release 1985-09-01
Pages 96
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Mathematics of Computing -- Parallelism.

Parallel Scientific Computing in C and MPI

Parallel Scientific Computing in C   and MPI Author George Em Karniadakis
ISBN-10 9781107494770
Release 2003-06-16
Pages 628
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Numerical algorithms, modern programming techniques, and parallel computing are often taught serially across different courses and different textbooks. The need to integrate concepts and tools usually comes only in employment or in research - after the courses are concluded - forcing the student to synthesise what is perceived to be three independent subfields into one. This book provides a seamless approach to stimulate the student simultaneously through the eyes of multiple disciplines, leading to enhanced understanding of scientific computing as a whole. The book includes both basic as well as advanced topics and places equal emphasis on the discretization of partial differential equations and on solvers. Some of the advanced topics include wavelets, high-order methods, non-symmetric systems, and parallelization of sparse systems. The material covered is suited to students from engineering, computer science, physics and mathematics.

Scientific Computing

Scientific Computing Author Gene H. Golub
ISBN-10 9781483296043
Release 2014-06-28
Pages 442
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This book introduces the basic concepts of parallel and vector computing in the context of an introduction to numerical methods. It contains chapters on parallel and vector matrix multiplication and solution of linear systems by direct and iterative methods. It is suitable for advanced undergraduate and beginning graduate courses in computer science, applied mathematics, and engineering. Ideally, students will have access to a parallel or Vector computer, but the material can be studied profitably in any case. * Gives a modern overview of scientific computing including parallel an vector computation * Introduces numerical methods for both ordinary and partial differential equations * Has considerable discussion of both direct and iterative methods for linear systems of equations, including parallel and vector algorithms * Covers most of the main topics for a first course in numerical methods and can serve as a text for this course

Numerical Solution of Partial Differential Equations

Numerical Solution of Partial Differential Equations Author K. W. Morton
ISBN-10 0521418550
Release 1994-10-20
Pages 239
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Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Generally these models must be solved numerically. This book provides a concise introduction to standard numerical techniques, ones chosen on the basis of their general utility for practical problems. The authors emphasise finite difference methods for simple examples of parabolic, hyperbolic and elliptic equations; finite element, finite volume and spectral methods are discussed briefly to see how they relate to the main theme. Stability is treated clearly and rigorously using maximum principles, energy methods, and discrete Fourier analysis. Methods are described in detail for simple problems, accompanied by typical graphical results. A key feature is the thorough analysis of the properties of these methods. Plenty of examples and exercises of varying difficulty are supplied. The book is based on the extensive teaching experience of the authors, who are also well-known for their work on practical and theoretical aspects of numerical analysis. It will be an excellent choice for students and teachers in mathematics, engineering and computer science departments seeking a concise introduction to the subject.

Numerical Solution of Partial Differential Equations by the Finite Element Method

Numerical Solution of Partial Differential Equations by the Finite Element Method Author Claes Johnson
ISBN-10 9780486131597
Release 2012-05-23
Pages 288
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An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.