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Code Verification Using The Method Of Manufactured Solutions
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Book Synopsis Code Verification Using The Method of Manufactured Solutions by :
Download or read book Code Verification Using The Method of Manufactured Solutions written by and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Implementations of numerical simulations for solving systems of partial differential equations are often not verified and are falsely assumed to work correctly. As a result, the implementations are prone to coding errors that could degrade the accuracy of the solution. In order to ensure that a code is written correctly, rigorous verification of all parts of the code is necessary. Code verification is the task of ascertaining whether a numerical algorithm is solving the governing equations of the problem correctly. If an exact solution existed for the governing equations then verification would be easier but these solutions are rare because of the non-linearity of common Computational Fluid Dynamics (CFD) problems. In the absence of exact solutions, grid refinement studies are the most commonly used methods to verify codes using simulations on a sequence of grids but even these studies have limitations. The Method of Manufactured Solutions (MMS) is a novel and a recently developed technique that verifies the observed order-of-accuracy of the implementation of a numerical algorithm. The method is more general and overcomes many of the limitations of the method of exact solutions and grid refinement studies. The central idea is to modify the governing equations and the boundary conditions by adding forcing functions or source terms in order to drive the discrete solution to a prescribed or ``manufactured' solution chosen a priori. A grid convergence study is performed subsequently to determine the observed orders. Two methods of accuracy assessment are presented here - solution accuracy analysis and residual error analysis. The method based on the error in the spatial residual is computationally less expensive and proved to be a valuable debugging tool. In the present work, the Method of Manufactured Solutions (MMS) is implemented on a compressible flow solver that solves the two-dimensional Euler equations on structured grids and an incompressible code that solves th.
Book Synopsis Code Verification Using the Method of Manufactured Solutions by : Vasanth Kumar Murali
Download or read book Code Verification Using the Method of Manufactured Solutions written by Vasanth Kumar Murali and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Implementations of numerical simulations for solving systems of partial differential equations are often not verified and are falsely assumed to work correctly. As a result, the implementations are prone to coding errors that could degrade the accuracy of the solution. In order to ensure that a code is written correctly, rigorous verification of all parts of the code is necessary. Code verification is the task of ascertaining whether a numerical algorithm is solving the governing equations of the problem correctly. If an exact solution existed for the governing equations then verification would be easier but these solutions are rare because of the non-linearity of common Computational Fluid Dynamics (CFD) problems. In the absence of exact solutions, grid refinement studies are the most commonly used methods to verify codes using simulations on a sequence of grids but even these studies have limitations. The Method of Manufactured Solutions (MMS) is a novel and a recently developed technique that verifies the observed order-of-accuracy of the implementation of a numerical algorithm. The method is more general and overcomes many of the limitations of the method of exact solutions and grid refinement studies. The central idea is to modify the governing equations and the boundary conditions by adding forcing functions or source terms in order to drive the discrete solution to a prescribed or 'manufactured' solution chosen a priori. A grid convergence study is performed subsequently to determine the observed orders. Two methods of accuracy assessment are presented here - solution accuracy analysis and residual error analysis. The method based on the error in the spatial residual is computationally less expensive and proved to be a valuable debugging tool. In the present work, the Method of Manufactured Solutions (MMS) is implemented on a compressible flow solver that solves the two-dimensional Euler equations on structured grids and an incompressible code that solves the two-dimensional Navier-Stokes equations on unstructured meshes. Exponential functions are used to 'manufacture' steady solutions to the governing equations. Solution and residual error analyses are presented. The influence of grid non-uniformity on the numerical accuracy is studied.
Book Synopsis Code Verification by the Method of Manufactured Solutions by :
Download or read book Code Verification by the Method of Manufactured Solutions written by and published by . This book was released on 2000 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: A procedure for code Verification by the Method of Manufactured Solutions (MMS) is presented. Although the procedure requires a certain amount of creativity and skill, we show that MMS can be applied to a variety of engineering codes which numerically solve partial differential equations. This is illustrated by detailed examples from computational fluid dynamics. The strength of the MMS procedure is that it can identify any coding mistake that affects the order-of-accuracy of the numerical method. A set of examples which use a blind-test protocol demonstrates the kinds of coding mistakes that can (and cannot) be exposed via the MMS code Verification procedure. The principle advantage of the MMS procedure over traditional methods of code Verification is that code capabilities are tested in full generality. The procedure thus results in a high degree of confidence that all coding mistakes which prevent the equations from being solved correctly have been identified.
Book Synopsis Computer Simulation Validation by : Claus Beisbart
Download or read book Computer Simulation Validation written by Claus Beisbart and published by Springer. This book was released on 2019-04-09 with total page 1056 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique volume introduces and discusses the methods of validating computer simulations in scientific research. The core concepts, strategies, and techniques of validation are explained by an international team of pre-eminent authorities, drawing on expertise from various fields ranging from engineering and the physical sciences to the social sciences and history. The work also offers new and original philosophical perspectives on the validation of simulations. Topics and features: introduces the fundamental concepts and principles related to the validation of computer simulations, and examines philosophical frameworks for thinking about validation; provides an overview of the various strategies and techniques available for validating simulations, as well as the preparatory steps that have to be taken prior to validation; describes commonly used reference points and mathematical frameworks applicable to simulation validation; reviews the legal prescriptions, and the administrative and procedural activities related to simulation validation; presents examples of best practice that demonstrate how methods of validation are applied in various disciplines and with different types of simulation models; covers important practical challenges faced by simulation scientists when applying validation methods and techniques; offers a selection of general philosophical reflections that explore the significance of validation from a broader perspective. This truly interdisciplinary handbook will appeal to a broad audience, from professional scientists spanning all natural and social sciences, to young scholars new to research with computer simulations. Philosophers of science, and methodologists seeking to increase their understanding of simulation validation, will also find much to benefit from in the text.
Book Synopsis Verification and Validation in Scientific Computing by : William L. Oberkampf
Download or read book Verification and Validation in Scientific Computing written by William L. Oberkampf and published by Cambridge University Press. This book was released on 2010-10-14 with total page 782 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.
Book Synopsis Verification of Computer Codes in Computational Science and Engineering by : Patrick Knupp
Download or read book Verification of Computer Codes in Computational Science and Engineering written by Patrick Knupp and published by CRC Press. This book was released on 2002-10-29 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: How can one be assured that computer codes that solve differential equations are correct? Standard practice using benchmark testing no longer provides full coverage because today's production codes solve more complex equations using more powerful algorithms. By verifying the order-of-accuracy of the numerical algorithm implemented in the code, one can detect most any coding mistake that would prevent correct solutions from being computed. Verification of Computer Codes in Computational Science and Engineering sets forth a powerful alternative called OVMSP: Order-Verification via the Manufactured Solution Procedure. This procedure has two primary components: using the Method of Manufactured Exact Solutions to create analytic solutions to the fully-general differential equations solved by the code and using grid convergence studies to confirm the order-of-accuracy. The authors present a step-by-step procedural guide to OVMSP implementation and demonstrate its effectiveness. Properly implemented, OVMSP offers an exciting opportunity to identify virtually all coding 'bugs' that prevent correct solution of the governing partial differential equations. Verification of Computer Codes in Computational Science and Engineering shows you how this can be done. The treatment is clear, concise, and suitable both for developers of production quality simulation software and as a reference for computational science and engineering professionals.
Book Synopsis Verification and Validation in Computational Science and Engineering by : Patrick J. Roache
Download or read book Verification and Validation in Computational Science and Engineering written by Patrick J. Roache and published by . This book was released on 1998 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Finite Difference Computing with PDEs by : Hans Petter Langtangen
Download or read book Finite Difference Computing with PDEs written by Hans Petter Langtangen and published by Springer. This book was released on 2017-06-21 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is open access under a CC BY 4.0 license. This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.
Book Synopsis Verification of a Compressible CFD Code Using the Method of Manufactured Solutions by : Christopher J. Roy
Download or read book Verification of a Compressible CFD Code Using the Method of Manufactured Solutions written by Christopher J. Roy and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Solving PDEs in Python by : Hans Petter Langtangen
Download or read book Solving PDEs in Python written by Hans Petter Langtangen and published by Springer. This book was released on 2017-03-21 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs. This book is open access under a CC BY license.
Book Synopsis Verification of 2D Navier-stokes Codes by the Method of Manufactured Solutions by : Vasanth Kumar Murali
Download or read book Verification of 2D Navier-stokes Codes by the Method of Manufactured Solutions written by Vasanth Kumar Murali and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Co-verification of Hardware and Software for ARM SoC Design by : Jason Andrews
Download or read book Co-verification of Hardware and Software for ARM SoC Design written by Jason Andrews and published by Elsevier. This book was released on 2004-09-04 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hardware/software co-verification is how to make sure that embedded system software works correctly with the hardware, and that the hardware has been properly designed to run the software successfully -before large sums are spent on prototypes or manufacturing. This is the first book to apply this verification technique to the rapidly growing field of embedded systems-on-a-chip(SoC). As traditional embedded system design evolves into single-chip design, embedded engineers must be armed with the necessary information to make educated decisions about which tools and methodology to deploy. SoC verification requires a mix of expertise from the disciplines of microprocessor and computer architecture, logic design and simulation, and C and Assembly language embedded software. Until now, the relevant information on how it all fits together has not been available. Andrews, a recognized expert, provides in-depth information about how co-verification really works, how to be successful using it, and pitfalls to avoid. He illustrates these concepts using concrete examples with the ARM core - a technology that has the dominant market share in embedded system product design. The companion CD-ROM contains all source code used in the design examples, a searchable e-book version, and useful design tools.* The only book on verification for systems-on-a-chip (SoC) on the market* Will save engineers and their companies time and money by showing them how to speed up the testing process, while still avoiding costly mistakes* Design examples use the ARM core, the dominant technology in SoC, and all the source code is included on the accompanying CD-Rom, so engineers can easily use it in their own designs
Book Synopsis VERIFICATION OF TRANSPORT CODES BY THE METHOD OF MANUFACTURED SOLUTIONS by :
Download or read book VERIFICATION OF TRANSPORT CODES BY THE METHOD OF MANUFACTURED SOLUTIONS written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We extend the Method of Manufactured Solutions (MMS) to the verification of transport codes. We derive analytic fixed sources required by the MMS procedure for several types of transport problems and apply the method to the Attila transport code. By means of this method we discover and correct several coding mistakes in Attila and ultimately verify its correct implementation for the problems studied. Our studies reveal that the MMS procedure is a useful tool for transport code development.
Author :American Bar Association. House of Delegates Publisher :American Bar Association ISBN 13 :9781590318737 Total Pages :216 pages Book Rating :4.3/5 (187 download)
Book Synopsis Model Rules of Professional Conduct by : American Bar Association. House of Delegates
Download or read book Model Rules of Professional Conduct written by American Bar Association. House of Delegates and published by American Bar Association. This book was released on 2007 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Model Rules of Professional Conduct provides an up-to-date resource for information on legal ethics. Federal, state and local courts in all jurisdictions look to the Rules for guidance in solving lawyer malpractice cases, disciplinary actions, disqualification issues, sanctions questions and much more. In this volume, black-letter Rules of Professional Conduct are followed by numbered Comments that explain each Rule's purpose and provide suggestions for its practical application. The Rules will help you identify proper conduct in a variety of given situations, review those instances where discretionary action is possible, and define the nature of the relationship between you and your clients, colleagues and the courts.
Book Synopsis Introduction to Applied Linear Algebra by : Stephen Boyd
Download or read book Introduction to Applied Linear Algebra written by Stephen Boyd and published by Cambridge University Press. This book was released on 2018-06-07 with total page 477 pages. Available in PDF, EPUB and Kindle. Book excerpt: A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.
Book Synopsis Thermal-Hydraulics of Water Cooled Nuclear Reactors by : Francesco D'Auria
Download or read book Thermal-Hydraulics of Water Cooled Nuclear Reactors written by Francesco D'Auria and published by Woodhead Publishing. This book was released on 2017-05-18 with total page 1200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thermal Hydraulics of Water-Cooled Nuclear Reactors reviews flow and heat transfer phenomena in nuclear systems and examines the critical contribution of this analysis to nuclear technology development. With a strong focus on system thermal hydraulics (SYS TH), the book provides a detailed, yet approachable, presentation of current approaches to reactor thermal hydraulic analysis, also considering the importance of this discipline for the design and operation of safe and efficient water-cooled and moderated reactors. Part One presents the background to nuclear thermal hydraulics, starting with a historical perspective, defining key terms, and considering thermal hydraulics requirements in nuclear technology. Part Two addresses the principles of thermodynamics and relevant target phenomena in nuclear systems. Next, the book focuses on nuclear thermal hydraulics modeling, covering the key areas of heat transfer and pressure drops, then moving on to an introduction to SYS TH and computational fluid dynamics codes. The final part of the book reviews the application of thermal hydraulics in nuclear technology, with chapters on V&V and uncertainty in SYS TH codes, the BEPU approach, and applications to new reactor design, plant lifetime extension, and accident analysis. This book is a valuable resource for academics, graduate students, and professionals studying the thermal hydraulic analysis of nuclear power plants and using SYS TH to demonstrate their safety and acceptability. - Contains a systematic and comprehensive review of current approaches to the thermal-hydraulic analysis of water-cooled and moderated nuclear reactors - Clearly presents the relationship between system level (top-down analysis) and component level phenomenology (bottom-up analysis) - Provides a strong focus on nuclear system thermal hydraulic (SYS TH) codes - Presents detailed coverage of the applications of thermal-hydraulics to demonstrate the safety and acceptability of nuclear power plants
Book Synopsis Introduction to Numerical Geodynamic Modelling by : Taras Gerya
Download or read book Introduction to Numerical Geodynamic Modelling written by Taras Gerya and published by Cambridge University Press. This book was released on 2010 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: This user-friendly reference for students and researchers presents the basic mathematical theory, before introducing modelling of key geodynamic processes.
