An Introduction to Numerical Methods and Analysis, 2nd Edition

CHAPTER 9

A SURVEY OF NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations (PDEs) are differential equations involving functions of more than one independent variable, such as the temperature at each point (x, y, z) in an iron bar. They are necessarily more complicated in some respects than ordinary differential equations, and it is beyond our intended scope here to provide an in-depth treatment of all the methods used for their approximate solution. Indeed, this is such an active research topic that it is best, in an introductory text, to confine ourselves to a brief survey of major ideas.

9.1 DIFFERENCE METHODS FOR THE DIFFUSION EQUATION

9.1.1 The Basic Problem

Perhaps the simplest PDE is the diffusion equation, so-called because it can be used to model a number of processes that are driven by diffusion, such as heat transfer and some types of slow mass transfer.

We seek the unknown function u(x, t) such that

(9.1)

(9.2)

(9.3)

(9.4)

Here g₀ and g₁ are the (known) boundary data, f, is a known source term, and u₀ is the (known) initial data. More general ...