CHAPTER 7
Finite volume methods for hyperbolic equations
1. Basic aspects
We have seen that the appearance of discontinuities even when starting from
smooth initial data is a generic situation for non-linear hyperbolic PDEs. To
dene what is meant by a sol
Canonical hyperbolic PDE systems
Linear equations
Elasticity
Consider a one-dimensional bar subject to a stretching force at one end and with the other end xed. The
force produces at time t > 0 a displacement d(x, t) of an innitesimal portion of the bar w
CHAPTER 11
Finite element methods
1. Preliminaries
For a number of applications the restrictions imposed by nite dierence or
spectral methods with respect to the computational grid are too severe. This is
especially the case in structural engineering wher
CHAPTER 1
Overview of frequently encountered PDEs
1. PDEs in the natural sciences
Ordinary and partial dierential equations (ODEs and PDEs henceforth) are
frequently encountered in numerous areas of study. A knowledge of the basic scientic background is n
CHAPTER 6
Finite dierence methods for hyperbolic equations
1. Scalar equations
1.1. Constant velocity advection in one dimension. The simplest example of a hyperbolic equation is the constant velocity advection equation
q t + u qx = 0
(1.1)
with some init
CHAPTER 9
Spectral methods
1. Preliminaries
We have so far used Fourier methods in the theoretical analysis of numerical
algorithms. However Fourier methods are also very useful in the construction of
numerical methods for PDEs. By way of an introduction
CHAPTER 2
Computational Fluid Dynamics
Historically, computational
uid dynamics (CFD) has been one of the rst
disciplines in which numerical methods have been applied widely. The main reason
underlying the early adoption of computational methods is the n
Chapter 1
TWODIMENSIONAL LAPLACES EQUATION
1.1
Introduction
Perhaps a good starting point for introducing boundary element methods is through
solving boundary value problems governed by the two-dimensional Laplaces equation
2 2
+
= 0.
x2 y 2
(1.1)
The La
CHAPTER 2
Numerical approaches to solving PDEs
1. A general framework for numerical solution of PDEs
Now that we have an idea of the dierential equations of interest in applications,
let us turn to the problem of nding solutions. Analytical techniques suc