Chapter 9
Solution Techniques for Elliptic
Problems
9.1 Direct Solution Methods
In Sections 8.1, 2, we saw that the discretization of an elliptic partial di erential equation
led to the solution of a large, sparse, linear algebraic system. In this chapter
Chapter 8
Elliptic Problems
8.1 Di erence Schemes for Poisson's Equation
Models of steady (time independent) problems in science and engineering give rise to
elliptic partial di erential systems. Examples arise in heat conduction, incompressible
ow, elect
Chapter 7
Multidimensional Hyperbolic
Problems
7.1 Split and Unsplit Di erence Methods
Our study of multidimensional parabolic problems in Chapter 5 has laid most of the
groundwork for our present task of creating di erence approximations for multidimensi
Chapter 6
Hyperbolic Problems
6.1 Vector Systems and Characteristics
Let's resume the study of nite di erence techniques for hyperbolic conservation laws by
examining the characteristics of one-dimensional vector systems having the form
ut + f (u)x = b(x
Chapter 5
Multi-Dimensional Parabolic
Problems
5.1 Alternating Direction Implicit (ADI) Methods
We would like to extend the one-dimensional explicit and implicit nite di erence schemes
that we have been studying to multi-dimensional parabolic problems. As
Chapter 4
Di erence Methods for Parabolic
Partial Di erential Equations
4.1 Implicit Di erence Methods
In this chapter, we focus on parabolic problems of which the now familiar one-dimensional
heat conduction problem
ut = uxx
0<x<1
u(x 0) = (x)
u(0 t) = f
Chapter 3
Basic Theoretical Concepts for
Time-Dependent Problems
3.1 Consistency, convergence, and stability
Three fundamental properties that every nite di erence approximation of a partial differential equation should possess are consistency, convergenc
Chapter 2
Introduction to Finite Di erence
Methods
2.1 Constructing Di erence Operators
Our initial study will involve the solution of time-dependent partial di erential equtions in
one space variable. We'll begin by introducing some elementary nite di er
Chapter 1
Introduction
1.1 Problems Leading to Partial Di erential Equations
Many problems in science and engineering are modeled and analyzed using systems of
partial di erential equations. Realistic models involve multi-dimensional behavior, nonlinearit