Vitoriano Ruas - An Introduction to Numerical Methods for Partial Differential Equations

1. Chapter 1
2. Chapter 5
3. Chapter 7

1. Chapter 1
2. Chapter 2
3. Chapter 3
4. Chapter 4
5. Chapter 5
6. Chapter 6
7. Chapter 7

Vitoriano Ruas

Sorbonne Universites, UPMC - Universite Paris 6, France

This edition first published 2016

2016 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Names: Ruas, Vitoriano, 1946- author.
Title: Numerical methods for partial differential equations : an introduction / Vitoriano Ruas.
Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, Inc., [2016] | Includes bibliographical references and index.
Identifiers: LCCN 2016002914 (print) | LCCN 2016007091 (ebook) | ISBN 9781119111351 (cloth) | ISBN 9781119111375 (pdf) | ISBN 9781119111368 (epub)
Subjects: LCSH: Differential equations, Partial. | Boundary value problems.
Classification: LCC QA374 .R83 2016 (print) | LCC QA374 (ebook) | DDC 515/.353dc23
LC record available at http://lccn.loc.gov/2016002914

A catalogue record for this book is available from the British Library.

A Alex, Lo & Romy

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Numerical methods have become nowadays indispensable tools for the quantitative solution of the differential equations that express the behaviour of any system in the universe. Examples are found in the study of engineering systems such as mechanical devices, structures and vehicles, and of biological and medical systems such as human organs, prosthesis and cells, to name a few. Numerical methods are applicable to the solution of differential equations that represent mathematical models of an underlying real system. These models are conceptual representation of reality using mathematics. It is interesting that in mechanics (which governs the behaviour of all engineering systems and many problems in physics) these models are obtained by simple balance (or equilibrium) laws applied to infinitesimal parts of the continuous system and its boundary. The beauty of the modelling process is that many times the resulting differential equations, even if they have the same mathematical form, are applicable to many different real life problems. A typical example is the Poisson equation that is used to represent the behaviour of such different problems as the heat conduction in a body, the seepage flow in a porous media, the flow of an incompressible fluid and the torsion of a bar, among others. Clearly a mathematical model represents a simplification of reality using geometrical data and physical properties of the constituent materials that are close enough to those of the real system under study and at the same time help to express its behaviour in mathematical form. On the other hand, the numerical solution of the resulting governing differential equations of the mathematical model introduces a series of simplifications. Numerical methods work on discretized forms of the model geometry which is split into simple geometrical entities, such as triangles and tetrahedra in two (2D) or three (3D) dimensions, respectively, as is typical in finite element or finite volume procedures or Cartesian grids, as is done in finite difference methods. The balance equations for each discretization unit (an element, a volume or a cell) are obtained in terms of a finite set of parameters and then used to obtain (assemble) the behaviour of the whole system by expressing the satisfaction of the balance laws at all the discrete points of the discrete system. The so-called discretization process leads to a system of algebraic equations that typically involve several millions of unknown parameters for real life problems. These equations are solved in modern computers using state of the art of computer science technology. This explains why numerical methods are many times referred to as computational methods. The two terms are in fact equivalent. The simplifications made along the modelling and numerical solution path lead to a final quantitative solution that is only approximate. The quantification of the modelling and numerical solution errors are topics of much current research. The timely book of Professor Ruas fits precisely in the context described above as it provides a comprehensive insight on the derivation and best use of numerical methods for solving a number of differential equations that govern a wide class of practical problems. The emphasis is in providing the reader with the knowledge and tools to assess the performance and reliability of a particular numerical method via the study of its stability, consistency, convergence and approximation order. The book uses simple one-dimensional (1D) problems for introducing the three basic numerical methods addressed in the book: finite differences, finite elements and finite volumes. A quantitative reliability analysis of the three families of methods is presented as well as the extension of the methods to transient situations and 2D problems. A convergence analysis for a well know multidimensional scalar problem (the Poisson equation) is also included. The book devotes a chapter to presenting higher order Lagrange finite elements and the extension of the finite element method to 3D problems. A final chapter describes a number of interesting topics such as the numerical solution of bi-harmonic equations in rectangles, the solution of the advection- diffusion equation, an outline of aposteriori error estimation and adaptive numerical solution procedures and a brief introduction to the numerical solution of nonlinear partial differential equations. Other numerical techniques are briefly presented in the Appendix.