Elements of Scientific Computing
Springer Science & Business Media, Sep 27, 2010 - Mathematics - 468 pages
Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.
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1 Computing Integrals
2 Differential Equations The First Steps
3 Systems of Ordinary Differential Equations
4 Nonlinear Algebraic Equations
5 The Method of Least Squares
6 About Scientific Software
7 The Diffusion Equation
8 Analysis of the Diffusion Equation
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Algorithm Algorithm 7.1 analytical solution apply approximation array assume boundary conditions called coefﬁcients compiler consider constant convergence deﬁned denote derive differential equation difﬁcult diffusion equation discrete efﬁcient error Euler scheme exact solution example Exercise explicit scheme exponential ﬁgure ﬁle ﬁnd ﬁnite difference ﬁrst ﬁsh ﬂow ﬂuid formula Fortran 77 Fourier Fourier sine series function f.x given graph heat conduction heun implementation implicit inﬁnite initial condition input interval Java least squares linear system loops Maple Matlab Newton’s method number of rabbits numerical integration numerical methods numerical solution object-oriented programming one-dimensional OpenMP parallel computing parameters physical plot processors programming language pseudo code Python quadratic result rnC1 satisﬁes scaling scientiﬁc computing sin.kx solve speciﬁc step sufﬁcient Taylor series temperature tion trapezoidal rule unC1 Unix values variable vector write zero