Numerical Partial Differential Equations: Finite Difference Methods (Texts in Applied Mathematics (22))

This is a book that approximates the solution of parabolic, first order hyperbolic and systems of partial differential equations using standard finite difference schemes (FDM). The theory and practice of FDM is discussed in detail and numerous practical examples (heat equation, convection-diffusion) in one and two space variables are given. In particular, Alternating Direction Implicit (ADI) methods are the standard means of solving PDE in 2 and 3 dimensions. In almost all cases model problems are taken in order to show how the schemes work for initial value problems, initial boundary value problem with Dirichlet and Neumann boundary conditions. This book is a *must* for those in science, engineering and quantitative financial analysis. It digs into the nitty-gritty of mapping a PDE to a FDM scheme while taking nasty boundary conditions into consideration. The resulting algorithms are documented are are easily programmed in C++ or other language. The book does not cover topics that are also important: operator splitting (Marchuk/Janenko), non-constant coefficient PDEs, nonlinearities. Finally, the book uses von Neumann analysis as a means of proving stability (getting a bit long in the tooth). There are more robust methods that use monotone schemes, M-matrices and the maximum principle. You should consult other specialised references. This is Volume I of a two-volume set (Volume II deals with Conversation Laws and first-order hyperbolic as well as Elliptic problems. (...) Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student in applied mathematics and engineering, this text offers a means of coming out of a course with a large number of methods that provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation. Prerequisites suggested for using this book in a course might include at least one semester of partial differential equations and some programming capability. The author stresses the use of technology throughout the text, allowing the student to utilize it as much as possible. The use of graphics for both illustration and analysis is emphasized, and algebraic manipulators are used when convenient. This is the second volume of a two-part book.

This text will be divided into two books which cover the topic of numerical partial differential equations. Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student, this text offers a means of coming out of a course with a large number of methods which provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation.

What makes this book stand out from the competition is that it is more computational. Once done with both volumes, readers will have the tools to attack a wider variety of problems than those worked out in the competitors' books. The author stresses the use of technology throughout the text, allowing students to utilize it as much as possible. Continuing the theme of the first, this second volume continues the study of the uses and techniques of numerical experimentation in the solution of PDE's. It includes topics such as initial-boundary-value problems, a complete survey of theory and numerical methods for conservation laws, and numerical schemes for elliptic PDE's. This chapter serves as an introduction to the subject of finite difference methods for solving partial differential equations. [1] Finite difference methods [2] Conservation laws and elliptic equations.