The Role Of Initial Value Solvers In A Moving Mesh Method

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The Role Of Initial Value Solvers In A Moving Mesh Method

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Author : Linda Ju
language : en
Publisher:
Release Date : 2007

The Role Of Initial Value Solvers In A Moving Mesh Method written by Linda Ju and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Differential-algebraic equations categories.

This thesis investigates the role of different initial value solvers in moving mesh software for solving second-order parabolic partial differential equations. We confine our attention primarily to the effect of the implicit ODE solver DDASSL used in the software MOVCOL of Huang and Russell, which is based on a moving collocation method. The advantages and limitations of Backward Differentiation Formula (BDF) methods and Implicit Runge-Kutta (IRK) methods are discussed in this particular context and the alternative ODE solver PSIDE, which is based on a four-stage RADAU IIA algorithm, is examined. The new interface of PSIDE with MOVCOL is discussed in some detail. Its performance is compared with DDASSL for several numerical problems.

Mesh Methods

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Author : Viktor A. Rukavishnikov
language : en
Publisher: MDPI
Release Date : 2021-03-29

Mesh Methods written by Viktor A. Rukavishnikov and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-03-29 with Mathematics categories.

Mathematical models of various natural processes are described by differential equations, systems of partial differential equations and integral equations. In most cases, the exact solution to such problems cannot be determined; therefore, one has to use grid methods to calculate an approximate solution using high-performance computing systems. These methods include the finite element method, the finite difference method, the finite volume method and combined methods. In this Special Issue, we bring to your attention works on theoretical studies of grid methods for approximation, stability and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. Of particular interest are new methods for solving boundary value problems with singularities, the complex geometry of the domain boundary and nonlinear equations. A part of the articles is devoted to the analysis of numerical methods developed for calculating mathematical models in various fields of applied science and engineering applications. As a rule, the ideas of symmetry are present in the design schemes and make the process harmonious and efficient.

Adaptive Moving Mesh Methods

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Author : Weizhang Huang
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-26

Adaptive Moving Mesh Methods written by Weizhang Huang 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 2010-10-26 with Mathematics categories.

This book is about adaptive mesh generation and moving mesh methods for the numerical solution of time-dependent partial differential equations. It presents a general framework and theory for adaptive mesh generation and gives a comprehensive treatment of moving mesh methods and their basic components, along with their application for a number of nontrivial physical problems. Many explicit examples with computed figures illustrate the various methods and the effects of parameter choices for those methods. Graduate students, researchers and practitioners working in this area will benefit from this book.

Partial Differential Equations

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Author : D. Sloan
language : en
Publisher: Elsevier
Release Date : 2012-12-02

Partial Differential Equations written by D. Sloan and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-02 with Mathematics categories.

/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight into the underlying stability and accuracy properties of computational algorithms for PDEs was deepened by building upon recent progress in mathematical analysis and in the theory of PDEs. To embark on a comprehensive review of the field of numerical analysis of partial differential equations within a single volume of this journal would have been an impossible task. Indeed, the 16 contributions included here, by some of the foremost world authorities in the subject, represent only a small sample of the major developments. We hope that these articles will, nevertheless, provide the reader with a stimulating glimpse into this diverse, exciting and important field. The opening paper by Thomée reviews the history of numerical analysis of PDEs, starting with the 1928 paper by Courant, Friedrichs and Lewy on the solution of problems of mathematical physics by means of finite differences. This excellent survey takes the reader through the development of finite differences for elliptic problems from the 1930s, and the intense study of finite differences for general initial value problems during the 1950s and 1960s. The formulation of the concept of stability is explored in the Lax equivalence theorem and the Kreiss matrix lemmas. Reference is made to the introduction of the finite element method by structural engineers, and a description is given of the subsequent development and mathematical analysis of the finite element method with piecewise polynomial approximating functions. The penultimate section of Thomée's survey deals with `other classes of approximation methods', and this covers methods such as collocation methods, spectral methods, finite volume methods and boundary integral methods. The final section is devoted to numerical linear algebra for elliptic problems. The next three papers, by Bialecki and Fairweather, Hesthaven and Gottlieb and Dahmen, describe, respectively, spline collocation methods, spectral methods and wavelet methods. The work by Bialecki and Fairweather is a comprehensive overview of orthogonal spline collocation from its first appearance to the latest mathematical developments and applications. The emphasis throughout is on problems in two space dimensions. The paper by Hesthaven and Gottlieb presents a review of Fourier and Chebyshev pseudospectral methods for the solution of hyperbolic PDEs. Particular emphasis is placed on the treatment of boundaries, stability of time discretisations, treatment of non-smooth solutions and multidomain techniques. The paper gives a clear view of the advances that have been made over the last decade in solving hyperbolic problems by means of spectral methods, but it shows that many critical issues remain open. The paper by Dahmen reviews the recent rapid growth in the use of wavelet methods for PDEs. The author focuses on the use of adaptivity, where significant successes have recently been achieved. He describes the potential weaknesses of wavelet methods as well as the perceived strengths, thus giving a balanced view that should encourage the study of wavelet methods.

Adaptive Mesh Methods And Software For Time Dependent Partial Differential Equations

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Author : Shengtai Li
language : en
Publisher:
Release Date : 1998

Adaptive Mesh Methods And Software For Time Dependent Partial Differential Equations written by Shengtai Li and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.

A Two Dimensional Mesh Moving Technique For Time Dependent Partial Differential Equations

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Author : D. C. Arney
language : en
Publisher:
Release Date : 1985

A Two Dimensional Mesh Moving Technique For Time Dependent Partial Differential Equations written by D. C. Arney and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with categories.

This document discusses an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of partial differential equations in two space dimensions. The mesh moving technique is based on an algebraic node movement function determined from the geometry and propagations of regions having significant discretization error indicators. This procedure is designed to be flexible, so that it can be used with many existing finite difference and finite element methods. To test the mesh moving algorithm, it was implemented in a system code with and initial mesh generator and a MacCormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws. Results are presented for several computational examples. The moving mesh scheme reduces dispersive errors near shocks and wave fronts and thereby reduces the grid requirements necessary to compute accurate solutions while incereasing computational efficiency. Additional keywords: Error clustering. (Author).

A Mesh Moving Technique For Time Dependent Partial Differential Equations In Two Space Dimensions

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Author : David C. Arney
language : en
Publisher:
Release Date : 198?

A Mesh Moving Technique For Time Dependent Partial Differential Equations In Two Space Dimensions written by David C. Arney and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 198? with Boundary value problems categories.

This article discusses an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of partial differential equations in two space dimensions and time. The mesh moving technique is based on an algebraic node movement function determined from the propagation of significant error regions. The algorithms is designed to be flexible, so that it can be used with many existing finite difference and finite element methods. To test the mesh moving algorithm, the authors implemented it in a system code with an initial mesh generator and a MacCormack finite volume scheme on quadralateral cells for hyperbolic vector systems. Results are presented for several computational examples. The moving mesh scheme reduces dispersion errors near shocks and wave fronts and thereby reduces the grid requirements necessary to compute accurate solutions while increasing computational efficiency.

Numerical Time Dependent Partial Differential Equations For Scientists And Engineers

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Author : Moysey Brio
language : en
Publisher: Academic Press
Release Date : 2010-09-21

Numerical Time Dependent Partial Differential Equations For Scientists And Engineers written by Moysey Brio and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-09-21 with Mathematics categories.

It is the first text that in addition to standard convergence theory treats other necessary ingredients for successful numerical simulations of physical systems encountered by every practitioner. The book is aimed at users with interests ranging from application modeling to numerical analysis and scientific software development. It is strongly influenced by the authors research in in space physics, electrical and optical engineering, applied mathematics, numerical analysis and professional software development. The material is based on a year-long graduate course taught at the University of Arizona since 1989. The book covers the first two-semesters of a three semester series. The second semester is based on a semester-long project, while the third semester requirement consists of a particular methods course in specific disciplines like computational fluid dynamics, finite element method in mechanical engineering, computational physics, biology, chemistry, photonics, etc. The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs; methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them. In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions; Practical stability analysis in the presence of the boundaries and interfaces; Treatment of problems with different temporal/spatial scales either explicit or implicit; preservation of symmetries and additional constraints; physical regularization of singularities; resolution enhancement using adaptive mesh refinement and moving meshes. Self contained presentation of key issues in successful numerical simulation Accessible to scientists and engineers with diverse background Provides analysis of the dispersion relation, symmetries, particular solutions and instabilities of the partial differential equations

Domain Decomposition Methods In Science And Engineering Xvii

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Author : Ulrich Langer
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-01-02

Domain Decomposition Methods In Science And Engineering Xvii written by Ulrich Langer 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 2008-01-02 with Mathematics categories.

Domain decomposition is an active, interdisciplinary research field concerned with the development, analysis, and implementation of coupling and decoupling strategies in mathematical and computational models. This volume contains selected papers presented at the 17th International Conference on Domain Decomposition Methods in Science and Engineering. It presents the newest domain decomposition techniques and examines their use in the modeling and simulation of complex problems.

An Adaptive Mesh Algorithm For Solving Systems Of Time Dependent Partial Differential Equations

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Author : David C. Arney
language : en
Publisher:
Release Date : 1985

An Adaptive Mesh Algorithm For Solving Systems Of Time Dependent Partial Differential Equations written by David C. Arney and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with categories.

This thesis discusses and adaptive mesh algorithm that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time dependent partial differential equations in two space dimensions. This algorithm combines the adaptive technique of mesh moving, static rezoning, and local mesh refinement. The nodes of a coarse mesh of quadrilateral cells are moved by a simple algebraic node movement function. The local mesh refinement method recursively divides cells of the moving coarse mesh within clustered regions that contain nodes with large error until a user prescribed error tolerance is satisfied. Keywords: Hyperbolic equations; Expert systems; and Computations.