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Open Access 2017 | Open Access | Buch

Finite Difference Computing with PDEs

A Modern Software Approach

verfasst von: Hans Petter Langtangen, Svein Linge

Verlag: Springer International Publishing

Buchreihe : Texts in Computational Science and Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis

Über dieses Buch

This book is open access under a CC BY 4.0 license.

This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

Inhaltsverzeichnis

Frontmatter

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1. Vibration ODEs

Abstract

Vibration problems lead to differential equations with solutions that oscillate in time, typically in a damped or undamped sinusoidal fashion. Such solutions put certain demands on the numerical methods compared to other phenomena whose solutions are monotone or very smooth. Both the frequency and amplitude of the oscillations need to be accurately handled by the numerical schemes. The forthcoming text presents a range of different methods, from classical ones (Runge-Kutta and midpoint/Crank-Nicolson methods), to more modern and popular symplectic (geometric) integration schemes (Leapfrog, Euler-Cromer, and Störmer-Verlet methods), but with a clear emphasis on the latter. Vibration problems occur throughout mechanics and physics, but the methods discussed in this text are also fundamental for constructing successful algorithms for partial differential equations of wave nature in multiple spatial dimensions.

Svein Linge, Hans Petter Langtangen

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2. Wave Equations

Abstract

A very wide range of physical processes lead to wave motion, where signals are propagated through a medium in space and time, normally with little or no permanent movement of the medium itself. The shape of the signals may undergo changes as they travel through matter, but usually not so much that the signals cannot be recognized at some later point in space and time. Many types of wave motion can be described by the equation $u_{tt}=\nabla\cdot(c^{2}\nabla u)+f$, which we will solve in the forthcoming text by finite difference methods.

Svein Linge, Hans Petter Langtangen

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3. Diffusion Equations

Abstract

The famous diffusion equation, also known as the heat equation, reads

$$\frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},$$

where $u(x,t)$ is the unknown function to be solved for, x is a coordinate in space, and t is time. The coefficient α is the diffusion coefficient and determines how fast u changes in time. A quick short form for the diffusion equation is $u_{t}=\alpha u_{xx}$.

Compared to the wave equation, $u_{tt}=c^{2}u_{xx}$, which looks very similar, the diffusion equation features solutions that are very different from those of the wave equation. Also, the diffusion equation makes quite different demands to the numerical methods.

Svein Linge, Hans Petter Langtangen

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4. Advection-Dominated Equations

Abstract

Wave (Chap. 2) and diffusion (Chap. 3) equations are solved reliably by finite difference methods. As soon as we add a first-order derivative in space, representing advective transport (also known as convective transport), the numerics gets more complicated and intuitively attractive methods no longer work well. We shall show how and why such methods fail and provide remedies. The present chapter builds on basic knowledge about finite difference methods for diffusion and wave equations, including the analysis by Fourier components, truncation error analysis (Appendix B), and compact difference notation.

Svein Linge, Hans Petter Langtangen

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5. Nonlinear Problems

Abstract

Algebraic equations

A linear, scalar, algebraic equation in x has the form

$$ax+b=0,$$

for arbitrary real constants a and b. The unknown is a number x. All other algebraic equations, e.g., $x^{2}+ax+b=0$, are nonlinear. The typical feature in a nonlinear algebraic equation is that the unknown appears in products with itself, like x ² or $e^{x}=1+x+\frac{1}{2}x^{2}+\frac{1}{3!}x^{3}+\ldots$

Svein Linge, Hans Petter Langtangen

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Backmatter

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Titel: Finite Difference Computing with PDEs
verfasst von: Hans Petter Langtangen
Svein Linge
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-55456-3
Print ISBN: 978-3-319-55455-6
DOI: https://doi.org/10.1007/978-3-319-55456-3

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Vibration ODEs

2. Wave Equations

3. Diffusion Equations

4. Advection-Dominated Equations

5. Nonlinear Problems

Backmatter

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