Johan Hoffman

Computational methods are an integral part of most scientific disciplines. This textbook introduces computational science through a set of methods and algorithms with the aim of familiarizing the reader with the field's theoretical foundations and providing the practical skills to use and develop computational methods.

Applied Mathematics: Body&Soul is a mathematics education reform p- gram including a series of books, together with associated educational ma- rial and open source software freely available from the project web page at www.bodysoulmath.org. Body&Soul re?ects the revolutionary new possibilities of mathematical modeling opened by the modern computer in the form of Computational C- culus (CC), which is now changing the paradigm of mathematical modeling in science and technology with new methods, questions and answers, as a modern form of the classical calculus of Leibniz and Newton. The Body&Soul series of books presents CC in a synthesis of com- tational mathematics (Body) and analytical mathematics (Soul) including applications. Volumes 1-3 [36] give a modern version of calculus and linear algebra including computation starting at a basic undergraduate level, and subsequent volumes on a graduate level cover di?erent areas of applications with focus on computational methods: • Volume 4: Computational Turbulent Incompressible Flow. • Volume 5: Computational Thermodynamics. • Volume 6: Computational Dynamical Systems. The present book is Volume 4, with Volumes 5 and 6 to appear in 2007 and further volumes on solid mechanics and electro-magnetics being planned. A gentle introduction to the Body&Soul series is given in [63].

Applied Mathematics: Body&Soul is a mathematics education reform p- gram including a series of books, together with associated educational ma- rial and open source software freely available from the project web page at www.bodysoulmath.org. Body&Soul re?ects the revolutionary new possibilities of mathematical modeling opened by the modern computer in the form of Computational C- culus (CC), which is now changing the paradigm of mathematical modeling in science and technology with new methods, questions and answers, as a modern form of the classical calculus of Leibniz and Newton. The Body&Soul series of books presents CC in a synthesis of com- tational mathematics (Body) and analytical mathematics (Soul) including applications. Volumes 1-3 [36] give a modern version of calculus and linear algebra including computation starting at a basic undergraduate level, and subsequent volumes on a graduate level cover di?erent areas of applications with focus on computational methods: • Volume 4: Computational Turbulent Incompressible Flow. • Volume 5: Computational Thermodynamics. • Volume 6: Computational Dynamical Systems. The present book is Volume 4, with Volumes 5 and 6 to appear in 2007 and further volumes on solid mechanics and electro-magnetics being planned. A gentle introduction to the Body&Soul series is given in [63].

The purpose of this book is to stimulate a much needed debate on ma- ematics education reform. We do not claim that we present anything near the whole truth, and that there are not many other points of view. We thus invite to debate and urge other people to express their views. As scientists we believe that it is our duty to present our own standpoints and conc- sions as clearly as possible, to open for scrutiny and discussion, rather than ambiguous politically correct views, which are di?cult to question. This book may be viewed as an introduction to our mathematics e- cation reform project Body&Soul based on a synthesis of Body (com- tational mathematics) and Soul (analytical mathematics). We do make a case for computational mathematics, which is the new world of mathem- ics opened by the computer, now waiting to enter mathematics education. But we do also propose a synthesis of this new world, with the kernel being computational algorithms, and the traditional world of analytical ma- ematics, with analytical formulas as the kernel. In fact, algorithms are expressed using analytical formulas before being translated into computer code, and good analytical formulas are necessary for understanding and insight.