High Order Difference Methods For Time Dependent Pde

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High Order Difference Methods For Time Dependent Pde

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Author : Bertil Gustafsson
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-12-06

High Order Difference Methods For Time Dependent Pde written by Bertil Gustafsson 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 2007-12-06 with Mathematics categories.

Many books have been written on ?nite difference methods (FDM), but there are good reasons to write still another one. The main reason is that even if higher order methods have been known for a long time, the analysis of stability, accuracy and effectiveness is missing to a large extent. For example, the de?nition of the formal high order accuracy is based on the assumption that the true solution is smooth, or expressed differently, that the grid is ?ne enough such that all variations in the solution are well resolved. In many applications, this assumption is not ful?lled, and then it is interesting to know if a high order method is still effective. Another problem that needs thorough analysis is the construction of boundary conditions such that both accuracy and stability is upheld. And ?nally, there has been quite a strongdevelopmentduringthe last years, inparticularwhenit comesto verygeneral and stable difference operators for application on initial–boundary value problems. The content of the book is not purely theoretical, neither is it a set of recipes for varioustypesof applications. The idea is to give an overviewof the basic theoryand constructionprinciplesfor differencemethodswithoutgoing into all details. For - ample, certain theorems are presented, but the proofs are in most cases left out. The explanation and application of the theory is illustrated by using simple model - amples.

Finite Difference Methods For Ordinary And Partial Differential Equations

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Author : Randall J. LeVeque
language : en
Publisher: SIAM
Release Date : 2007-01-01

Finite Difference Methods For Ordinary And Partial Differential Equations written by Randall J. LeVeque and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.

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.

Spectral And High Order Methods For Partial Differential Equations Icosahom 2014

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Author : Robert M. Kirby
language : en
Publisher: Springer
Release Date : 2015-11-26

Spectral And High Order Methods For Partial Differential Equations Icosahom 2014 written by Robert M. Kirby and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-26 with Computers categories.

The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2014), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of papers will provide the reader with a snapshot of the state-of-the-art and help initiate new research directions through the extensive biography.

Spectral And High Order Methods For Partial Differential Equations

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Author : Jan S. Hesthaven
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-29

Spectral And High Order Methods For Partial Differential Equations written by Jan S. Hesthaven 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 2010-10-29 with Mathematics categories.

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.

Time Dependent Problems And Difference Methods

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Author : Bertil Gustafsson
language : en
Publisher: John Wiley & Sons
Release Date : 2013-07-18

Time Dependent Problems And Difference Methods written by Bertil Gustafsson and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-18 with Mathematics categories.

Praise for the First Edition ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations." —SIAM Review Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.

Efficient High Order Discretizations For Computational Fluid Dynamics

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Author : Martin Kronbichler
language : en
Publisher: Springer Nature
Release Date : 2021-01-04

Efficient High Order Discretizations For Computational Fluid Dynamics written by Martin Kronbichler and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-04 with Technology & Engineering categories.

The book introduces modern high-order methods for computational fluid dynamics. As compared to low order finite volumes predominant in today's production codes, higher order discretizations significantly reduce dispersion errors, the main source of error in long-time simulations of flow at higher Reynolds numbers. A major goal of this book is to teach the basics of the discontinuous Galerkin (DG) method in terms of its finite volume and finite element ingredients. It also discusses the computational efficiency of high-order methods versus state-of-the-art low order methods in the finite difference context, given that accuracy requirements in engineering are often not overly strict. The book mainly addresses researchers and doctoral students in engineering, applied mathematics, physics and high-performance computing with a strong interest in the interdisciplinary aspects of computational fluid dynamics. It is also well-suited for practicing computational engineers who would like to gain an overview of discontinuous Galerkin methods, modern algorithmic realizations, and high-performance implementations.

Finite Difference Methods For Compressible Two Fluid Dynamics

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Author : Khosro Shahbazi
language : en
Publisher: Springer Nature
Release Date : 2025-07-04

Finite Difference Methods For Compressible Two Fluid Dynamics written by Khosro Shahbazi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-04 with Mathematics categories.

Finite Difference Methods for Compressible Two-Fluid Dynamics provides the essentials of high-order numerical methods for compressible single-fluid and two-fluid transport phenomena. This book can serve as a first course on the numerical methods for transport phenomena or fluid dynamics for students in mechanical, aerospace, and chemical engineering, applied mathematics, and physics at the senior level of an undergraduate or graduate degree. It also provides foundations and algorithmic details for implementing the most recent numerical schemes for compressible flows and extending them to include other physics, such as elasticity, reaction, and magnetohydrodynamics. The book's presented schemes enable computations for broad applications, including shock-induced interfacial instability and turbulence, shock-bubble interactions, and detonation, to name a few. For a broad reach and impact, the numerical schemes satisfy the simultaneous requirements of simplicity, extendibility, and efficiency on serial and parallel computers. The physics of the compressible single- and two-fluid system also guide the design and analysis of the numerical methods. The enabled direct numerical simulations also help obtain accurate data for tuning the emerging physics-based neuromorphic algorithms.

Spectral And High Order Methods For Partial Differential Equations Icosahom 2016

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Author : Marco L. Bittencourt
language : en
Publisher: Springer
Release Date : 2017-11-07

Spectral And High Order Methods For Partial Differential Equations Icosahom 2016 written by Marco L. Bittencourt and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-07 with Mathematics categories.

This book features a selection of high-quality papers chosen from the best presentations at the International Conference on Spectral and High-Order Methods (2016), offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions.

High Order Finite Difference Approximations For Hyperbolic Problems

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Author : Hannes Frenander
language : en
Publisher: Linköping University Electronic Press
Release Date : 2017-01-24

High Order Finite Difference Approximations For Hyperbolic Problems written by Hannes Frenander and has been published by Linköping University Electronic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-24 with categories.

In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.

Analytic Methods For Partial Differential Equations

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Author : G. Evans
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Analytic Methods For Partial Differential Equations written by G. Evans 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 2012-12-06 with Mathematics categories.

The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. J ames Clerk Maxwell, for example, put electricity and magnetism into a unified theory by estab lishing Maxwell's equations for electromagnetic theory, which gave solutions for problems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechankal processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier-Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forcasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics.