This is a book about numerically solving partial differential equations occurring in technical and physical contexts and we (the authors) have set ourselves a more ambitious target than to just talk about the numerics. Our aim is to show the place of numerical solutions in the general modeling process and this must inevitably lead to considerations about modeling itself. Partial differential equations usually are a consequence of applying first principles to a technical or physical problem at hand. That means, that most of the time the physics also have to be taken into account especially for validation of the numerical solution obtained.
This book in other words is especially aimed at engineers and scientists who have 'realworld' problems and itwill concern itself lesswith peskymathematical detail.
For the interested reader though, we have included sections on mathematical theory to provide the necessary mathematical background. Since this treatment had to be on the superficial side we have provided further reference to the literature where necessary.
From the improved edition of 2008 exercises and theory are more separately presented.
Furthermore, some parts, such as the parts on boundary fitted coordinates, on
coordinate transformation, the treatment of essential boundary conditions for FEM
and the solution of non-linear systems of equations, have been rewritten to make
them easier to understand. Newmark-type solvers for the wave equation have been added.
In this improved second edition, 2014, the treatment of boundary conditions for all types
of discretizationmethods has been extended. Periodical boundary conditions have
been included. Furthermore,the description of the FEM has been simplified.
1 Review of some basic mathematical concepts
2 A crash course in PDE's
3 Finite difference methods
4 Finite volume methods
5 Minimization problems in physics
6 The numerical solution of minimization problems
7 The weak formulation and Galerkin's method
8 Extension of the FEM
9 Solution of large systems of equations
10 The heat- or diffusion equation
11 The wave equation
12 The transport equation
13 Moving boundary problems