Diffpack Online

Advanced Topics in Computational PDEs

Numerical Methods and Diffpack Programming

Langtangen, Hans P.; Tveito, Aslak (Eds.)

[Back-cover text] [Excerpt from Preface] [Table of Contents]

Back-cover text

The book is suitable for readers with a background in basic finite element and finite difference methods for partial differential equations who wants gentle introductions to advanced topics like parallel computing, multigrid methods, and special methods for systems of PDEs. The goal of all chapters is to 'compute' solutions to problems, hence algorithmic and software issues play a central role. All software examples use the Diffpack programming environment, so to take advantage of these examples some experience with Diffpack is required. There are also some chapters covering complete applications, i.e., the way from a model, expressed as systems of PDEs, through discretization methods, algorithms, software design, verification, and computational examples.

Excerpt from Preface

This book is about solving partial differential equations (PDEs). Such equations are used to model a wide range of phenomena in virtually all fields of science and technology. In the last decade, the general availability of extremely powerful computers has shifted the focus in computational mathematics from simplified model problems to much more sophisticated models resembling intricate features of real life. This change challenges our knowledge in computer science and in numerical analysis.
The main objective of the present book is to teach modern, advanced techniques for numerical PDE solution. The book also introduces several models arising in fields like finance, medicine, material technology, and geology. In order to read this book, you must have a basic knowledge of partial differential equations and numerical methods for solving such equations. Furthermore, some background in finite element methods is required. You do not need to know Diffpack, although this programming environment is used in examples throughout the text. Basically, this book is about models, methods, and how to implement the methods. For the implementation part it is natural for us to use Diffpack as the programming environment, because making a PDE solver in Diffpack requires little amount of programming and because Diffpack has support for the advanced numerical methods treated in this book. Most chapters have a part on models and methods, and a part on implementation and Diffpack programming. The exposition is designed such that readers can focus only on the first part, if desired. The prerequisites for the present book (PDEs and numerics) as well as an introduction to Diffpack programming can be found in "Computational Partial Differential Equations - Numerical Methods and Diffpack Programming" by H. P. Langtangen.

Table of Contents


X. Cai, E. Acklam, H. P. Langtangen, A. Tveito: Parallel Computing.
X. Cai: Overlapping Domain Decomposition Methods.
K.-A. Mardal, G. W. Zumbusch, H. P. Langtangen: Software Tools for Multigrid Methods.
K.-A. Mardal, H. P. Langtangen: Mixed Finite Elements.
K.-A. Mardal, J. Sundnes, H. P. Langtangen, A. Tveito: Systems of PDEs and Block Preconditioning.
Å Ødegård, H. P. Langtangen, A. Tveito: Object-Oriented Implementation of Fully Implicit Methods for Systems of PDEs.
H. P. Langtangen, H. Osnes: Stochastic Partial Differential Equations.
H. P. Langtangen and K.-A. Mardal: Using Diffpack from Python Scripts.
X. Cai, A. M. Bruaset, H. P. Langtangen, G. T. Lines, K. Samuelsson, W. Shen, A. Tveito, G. Zumbusch: Performance Modeling of PDE Solvers.
J. Sundnes, G.T. Lines, P. Grøttum, A. Tveito: Numerical Methods and Software for Modeling the Electrical Activity in the Human Heart.
O. Skavhaug, B. F. Nielsen, A. Tveito: Mathematical Models of Financial Derivatives.
O. Skavhaug, B. F. Nielsen, A. Tveito: Numerical Methods for Financial Derivatives.
T. Thorvaldsen, H. P. Langtangen, H. Osnes: Finite Element Modeling of Elastic Structures.
K. M. Okstad, T. Kvamsdal: Simulation of Aluminum Extrusion.
A. Kjeldstad, H. P. Langtangen, J. Skogseid, K. Bjørlykke: Simulation of Deformations, Fluid Flow and Heat Transfer in Sedimentary Basins