Benjamin A. Stickler - Basic Concepts in Computational Physics

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 differentiation and integration, the treatment of ordinary differential equations, and we present some notes on the numerics of partial differential equations. Each section 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 following 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 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 analysis of the various topics and to even provide in some cases the necessary means to understand the very background of these methods. Although there is a barely comprehensible 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 par with stochastic methods and to motivate these methods by concrete examples out of physics and/or engineering.

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 and the stochastic methods, we created a stimulating presentation of the basic concepts applied in Computational Physics.

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.