The book of nature is written in the language of mathematics -- Galileo Galilei
How is it possible to predict weather patterns for tomorrow, with access solely to today’s weather data? And how is it possible to predict the aerodynamic behavior of an aircraft that has yet to be built?
The answer is computer simulations based on mathematical models – sets of equations – that describe the underlying physical properties. However, these equations are usually much too complicated to solve, either by the smartest mathematician or the largest supercomputer. This problem is overcome by constructing an approximation: a numerical model with a simpler structure can be translated into a program that tells the computer how to carry out the simulation.
This book conveys the fundamentals of mathematical models, numerical methods and algorithms. Opening with a tutorial on mathematical models and analysis, it proceeds to introduce the most important classes of numerical methods, with finite element, finite difference and spectral methods as central tools. The concluding section describes applications in physics and engineering, including wave propagation, heat conduction and fluid dynamics. Also covered are the principles of computers and programming, including MATLAB®.
Author(s): Bertil Gustafsson (auth.)
Series: Texts in Computational Science and Engineering 8
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2011
Language: English
Pages: 326
Tags: Computational Mathematics and Numerical Analysis;Computational Science and Engineering;Computer Appl. in Life Sciences;Appl.Mathematics/Computational Methods of Engineering;Numerical and Computational Physics;Simulation and Modeling
Front Matter....Pages I-XIII
Front Matter....Pages 1-1
Introduction....Pages 3-17
Mathematical Models....Pages 19-45
Basic Linear Algebra....Pages 47-65
Analysis Tools....Pages 67-75
Elementary Functions....Pages 77-99
Front Matter....Pages 101-101
The Fourier Transform....Pages 103-115
Polynomial Expansions....Pages 117-123
Least Square Problems....Pages 125-133
Front Matter....Pages 135-135
Numerical Methods for Differential Equations....Pages 137-144
Finite Difference Methods....Pages 145-171
Finite Element Methods....Pages 173-192
Spectral Methods....Pages 193-202
Front Matter....Pages 203-203
Numerical Solution of Nonlinear Equations....Pages 205-211
Linear Systems of Equations....Pages 213-231
Front Matter....Pages 233-233
Wave Propagation....Pages 235-254
Heat Conduction....Pages 255-262
Fluid Dynamics....Pages 263-289
Computers and Programming....Pages 291-302
Further Reading....Pages 303-304
Erratum to: Basic Linear Algebra....Pages E1-E1
Back Matter....Pages 305-322
