Basic Concepts in Computational Physics Benjamin A. Stickler • Ewald Schachinger Basic Concepts in Computational Physics 123 Benjamin A. Stickler Ewald Schachinger Faculty of Physics Institute of Theoretical and Computational University of Duisburg-Essen Physics Duisburg Graz University of Technology Germany Graz Austria ISBN 978-3-319-02434-9 ISBN 978-3-319-02435-6 (eBook) DOI 10.1007/978-3-319-02435-6 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013949666 Ó Springer International Publishing Switzerland 2014 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Traditionally physics was divided into two fields of activities: theoretical and experimental. As a consequence of the stunning increase in computer power and of the development of more powerful numerical techniques, a new branch of physics was established over the last decades: Computational Physics. This new branch introduced as a spin-off what nowadays is commonly called computer simulations, which play an increasingly important role in physics and in related sciences as well as in industrial applications. They serve two purposes • Direct simulation of physical processes (theoretical-experimental-physics) such as for instance – molecular dynamics or – Monte Carlo simulation of physical processes. • Solution of complex mathematical problems such as – differential equations, – minimization problems as well as – high dimensional integrals or sums. This book addresses all these scenarios on a very basic level. It is addressed to lecturers who will have to teach a basic course/basic courses in Computational Physics or Numerical Methods and to students as a companion in their first steps into the realm of this fascinating field of modern research. Following these intentions, this book was divided into two parts. Part I deals with deterministic methods in Computational Physics. We discuss, in particular, numerical differ- entiation and integration, the treatment of ordinary differential equations, and we present some notes on the numerics of partial differential equations. Each chapter within this part of the book is complemented by numerous applications. Part II of this book provides an introduction to stochastic methods in Computational Physics. In particular, we will examine how to generate random numbers fol- lowing a given distribution, summarize the basics of stochastics in order to establish the necessary background to understand techniques like MARKOV-Chain Monte Carlo. Finally, algorithms of stochastic optimization are discussed. Again, numerous examples out of physics like diffusion processes or the POTTS model are v vi Preface investigated exhaustively. Finally, this book contains an Appendix that augments the main parts of the book with a detailed discussion of supplementary topics. This book is not meant to be just a collection of algorithms which can immediately be applied to various problems which may arise in Computational Physics. On the contrary, the scope of this book is to provide the reader with a mathematically well founded glance behind the scene of Computational Physics. Thus, particular emphasis is on a clear mathematical analysis of the various topics and to even provide, in some cases, the necessary mathematical means to under- stand the very background of these methods. Although there is a barely compre- hensible amount of excellent literature on Computational Physics, most of these books seem to concentrate either on deterministic methods or on stochastic methods. It is not our goal to compete with these rather specific works. On the contrary, it is the particular focus of this book to discuss deterministic methods on a par with stochastic methods. Nevertheless, a certain overlap with existing literature was unavoidable and we apologize if we were not able to cite appropriately all existing works which are of importance and which influenced this book. However, we believe that by putting the emphasis on an exact mathematical analysis of both the deterministic as well as the stochastic methods we created a stimulating presentation of the basic concepts applied in Computational Physics. If we assume two basic courses in Computational Physics to be part of the curriculum, nick-named here: Computational Physics 101 and Computational Physics 102, then we would like to suggest to present/study the various topics of this book according to the following syllabus: • Computational Physics 101: – Chapter 1: Some Basic Remarks. – Chapter 2: Numerical Differentiation. – Chapter 3: Numerical Integration. – Chapter 4:TheKEPLER Problem. – Chapter 5: Ordinary Differential Equations: Initial Value Problems. –(Chapter 6: The Double Pendulum). – Chapter 7: Molecular Dynamics. – Chapter 8: Numerics of Ordinary Differential Equations: Boundary Value Problems. –(Chapter 9: The One-Dimensional Stationary Heat Equation). – Chapter 10: The One-Dimensional Stationary SCHRÖDINGER Equation. – Chapter 12: Pseudo Random Number Generators. • Computational Physics 102: – Chapter 11: Partial Differential Equations. – Chapter 13: Random Sampling Methods. – Chapter 14: A Brief Introduction to Monte-Carlo Methods. – Chapter 15:TheISING Model. Preface vii – Chapter 16: Some Basics of Stochastic Processes. – Chapter 17: The Random Walk and Diffusion Theory. – Chapter 18:MARKOV-Chain Monte Carlo and the POTTS Model. – Chapter 19: Data Analysis. – Chapter 20: Stochastic Optimization. The various chapters are augmented by problems of medium complexity which help to understand better the numerical part of the topics discussed within this book. Although the manuscript has been carefully checked several times, we cannot exclude that some errors escaped our scrutiny. We apologize in advance and would highly appreciate reports of potential mistakes or typos to Benjamin A. Stickler ([email protected]). Throughout the book SI-units are used except stated otherwise. Graz, July 2013 Benjamin A. Stickler Ewald Schachinger Acknowledgments The Authors are grateful to Prof. Dr. C. Gattringer (Karl-Franzens Universität, Graz), to Profs. Dr. W. von der Linden, Dr. H.-G. Evertz, and Dr. H. Sormann (Graz University of Technology, Austria). They inspired this book with their lectures on various topics of Computational Physics, computer simulation, and numerical analysis. Last but not least the Authors thank Dr. Chris Theis (CERN) for meticulously reading the manuscript, for pointing out inconsistencies, and for suggestions to improve the text. ix Contents 1 Some Basic Remarks ................................. 1 1.1 Motivation . 1 1.2 Rounding Errors . 6 1.3 Methodological Errors. 8 1.4 Stability. 9 1.5 Concluding Remarks. 12 References . 13 Part I Deterministic Methods 2 Numerical Differentiation .............................. 17 2.1 Introduction . 17 2.2 Finite Differences . 18 2.3 Finite Difference Derivatives . 20 2.4 A Systematic Approach: The Operator Technique . 22 2.5 Concluding Discussion . 25 Summary . 27 Problems. 28 References . 28 3 Numerical Integration ................................ 29 3.1 Introduction . 29 3.2 Rectangular Rule . 30 3.3 Trapezoidal Rule . 33 3.4 The SIMPSON Rule . 35 3.5 General Formulation: The NEWTON-COTES Rules . 36 3.6 GAUSS-LEGENDRE Quadrature. 38 3.7 An Example. 44 3.8 Concluding Discussion . 45 Summary . 47 Problems. 48 References . 49 xi xii Contents 4 The KEPLER Problem ................................. 51 4.1 Introduction . 51 4.2 The Problem . 52 4.3 Numerical Treatment . 54 Summary . 59 References . 59 5 Ordinary Differential Equations: Initial Value Problems ....... 61 5.1 Introduction . 61 5.2 Simple Integrators . 62 5.3 RUNGE-KUTTA Methods . 66 5.4 Hamiltonian Systems: Symplectic Integrators. 71 5.5 An Example: The KEPLER Problem, Revisited . 73 Summary