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Macmillan Higher Education Palgrave Higher Education

Basic Concepts in Computational Physics

ISBN 9783319024356
Publication Date December 2013
Formats Ebook 
Publisher Springer

With the development of ever more powerful computers a new branch of physics and engineering evolved over the last few decades: Computer Simulation or Computational Physics. It serves two main purposes:
- Solution of complex mathematical problems such as, differential equations, minimization/optimization, or high-dimensional sums/integrals.
- Direct simulation of physical processes, as for instance, molecular dynamics or Monte-Carlo simulation of physical/chemical/technical processes.
Consequently, the book is divided into two main parts: Deterministic methods and stochastic methods. Based on concrete problems, the first part discusses numerical differentiation and integration, and the treatment of ordinary differential equations. This is augmented by notes on the numerics of partial differential equations. The second part discusses the generation of random numbers, summarizes the basics of stochastics which is then followed by the introduction of various Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. All this is again augmented by numerous applications from physics. The final two chapters on Data Analysis and Stochastic Optimization share the two main topics as a common denominator. The book offers a number of appendices to provide the reader with more detailed information on various topics discussed in the main part. Nevertheless, the reader should be familiar with the most important concepts of statistics and probability theory albeit two appendices have been dedicated to provide a rudimentary discussion.

Ewald Schachinger
Institut für Theoretische und Computational Physik,
Technische Universität Graz, Petersgasse 16, A-8010 Graz
Benjamin A. Stickler
Institut für Theoretische Physik, Karl Franzens Universität
Graz, Universitätsplatz 5, A-8010 Graz, benjamin.stickler@uni-graz.at

Part I Deterministic Methods
Numerical Differentiation
Numerical Integration
The KEPLER Problem
Ordinary Differential Equations – Initial Value Problem
The Double Pendulum
Molecular Dynamics
Numerics of Ordinary Differential Equations – Boundary Value Problems
The One-Dimensional Heat Equation
The One-Dimensional SCHRÖDINGER Equation
Introduction to the Numerics of Partial Differential Equations
Part II Stochastic Methods
Pseudo Random Number Generation
Random Sampling Methods
A Brief Introduction to Monte-Carlo Methods
The ISING Model
Some Basics of Stochastic Processes
The Random Walk and Diffusion Theory
MARKOV Chain Monte Carlo and the POTTS Model
Data Analysis
Stochastic Optimization
Part III Appendix
Solving Non-Linear Equations. The NEWTON Method
Numerical Solution of Systems of Linear Equations
Basics of Probability Theory
Phase Transitions
Fractional Integrals and Derivatives in One Dimension,- Least Squares Fit
Deterministic Optimization


From the reviews:“The authors characterize the aim of their book to ‘address the scenarios of direct simulation of physical processes and the solution of complex mathematical problems on a very basic level’. It is directed to lecturers teaching basic courses in Computational Physics and to students as a companion when starting studying in this field.” (Rolf Dieter Grigorieff, zbMATH, Vol. 1287, 2014)
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