Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion

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Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion Equations

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Author : Donald J. Estep
language : en
Publisher: American Mathematical Soc.
Release Date : 2000

Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion Equations written by Donald J. Estep and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.

Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion

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Author : Donald J. Estep
language : en
Publisher:
Release Date : 2014-09-11

Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion written by Donald J. Estep and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-11 with Error analysis categories.

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. The authors outline, analyze and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally.

Journal Of Numerical Mathematics

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Author :
language : en
Publisher:
Release Date : 2004

Journal Of Numerical Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Numerical analysis categories.

Siam Journal On Scientific Computing

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Author :
language : en
Publisher:
Release Date : 2008

Siam Journal On Scientific Computing written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematical statistics categories.

Multiscale Methods

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Author : Jacob Fish
language : en
Publisher: Oxford University Press
Release Date : 2010

Multiscale Methods written by Jacob Fish and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.

Small scale features and processes occurring at nanometer and femtosecond scales have a profound impact on what happens at a larger scale and over an extensive period of time. The primary objective of this volume is to reflect the state-of-the-art in multiscale mathematics, modeling, and simulations and to address the following barriers: What is the information that needs to be transferred from one model or scale to another and what physical principles must be satisfied during thetransfer of information? What are the optimal ways to achieve such transfer of information? How can variability of physical parameters at multiple scales be quantified and how can it be accounted for to ensure design robustness?The multiscale approaches in space and time presented in this volume are grouped into two main categories: information-passing and concurrent. In the concurrent approaches various scales are simultaneously resolved, whereas in the information-passing methods the fine scale is modeled and its gross response is infused into the continuum scale. The issue of reliability of multiscale modeling and simulation tools which focus on a hierarchy of multiscale models and an a posteriori model of errorestimation including uncertainty quantification, is discussed in several chapters. Component software that can be effectively combined to address a wide range of multiscale simulations is also described. Applications range from advanced materials to nanoelectromechanical systems (NEMS), biologicalsystems, and nanoporous catalysts where physical phenomena operates across 12 orders of magnitude in time scales and 10 orders of magnitude in spatial scales.This volume is a valuable reference book for scientists, engineers and graduate students practicing in traditional engineering and science disciplines as well as in emerging fields of nanotechnology, biotechnology, microelectronics and energy.

Verification And Validation In Scientific Computing

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Author : William L. Oberkampf
language : en
Publisher: Cambridge University Press
Release Date : 2010-10-14

Verification And Validation In Scientific Computing written by William L. Oberkampf and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-14 with Computers categories.

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.

Applied Mechanics Reviews

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Author :
language : en
Publisher:
Release Date : 1977

Applied Mechanics Reviews written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1977 with Mechanics, Applied categories.

Error Estimation And Adaptive Discretization Methods In Computational Fluid Dynamics

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Author : Timothy J. Barth
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Error Estimation And Adaptive Discretization Methods In Computational Fluid Dynamics written by Timothy J. Barth and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

As computational fluid dynamics (CFD) is applied to ever more demanding fluid flow problems, the ability to compute numerical fluid flow solutions to a user specified tolerance as well as the ability to quantify the accuracy of an existing numerical solution are seen as essential ingredients in robust numerical simulation. Although the task of accurate error estimation for the nonlinear equations of CFD seems a daunting problem, considerable effort has centered on this challenge in recent years with notable progress being made by the use of advanced error estimation techniques and adaptive discretization methods. To address this important topic, a special course wasjointly organized by the NATO Research and Technology Office (RTO), the von Karman Insti tute for Fluid Dynamics, and the NASA Ames Research Center. The NATO RTO sponsored course entitled "Error Estimation and Solution Adaptive Discretization in CFD" was held September 10-14, 2002 at the NASA Ames Research Center and October 15-19, 2002 at the von Karman Institute in Belgium. During the special course, a series of comprehensive lectures by leading experts discussed recent advances and technical progress in the area of numerical error estimation and adaptive discretization methods with spe cific emphasis on computational fluid dynamics. The lecture notes provided in this volume are derived from the special course material. The volume con sists of 6 articles prepared by the special course lecturers.

Cellular Automata

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Author : Hiroshi Umeo
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-08-28

Cellular Automata written by Hiroshi Umeo and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-08-28 with Computers categories.

This book constitutes the refereed proceedings of the 8th International Conference on Cellular Automata for Research and Industry, ACRI 2008, held in Yokohama, Japan, in September 2008. The 43 revised full papers and 22 revised poster papers presented together with 4 invited lectures were carefully reviewed and selected from 78 submissions. The papers focus on challenging problems and new research not only in theoretical but application aspects of cellular automata, including cellular automata tools and computational sciences. The volume also contains 11 extended abstracts dealing with crowds and cellular automata, which were presented during the workshop C&CA 2008. The papers are organized in topical sections on CA theory and implementation, computational theory, physical modeling, urban, environmental and social modeling, pedestrian and traffic flow modeling, crypto and security, system biology, CA-based hardware, as well as crowds and cellular automata.

Boundary And Interior Layers Computational And Asymptotic Methods Bail 2016

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Author : Zhongyi Huang
language : en
Publisher: Springer
Release Date : 2017-10-26

Boundary And Interior Layers Computational And Asymptotic Methods Bail 2016 written by Zhongyi Huang and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-26 with Mathematics categories.

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.