ABSTRACT
In this chapter, we examine in detail various numerical schemes that can be used to solve simple model partial differential equations (PDEs). These model equations include the ƒrst-order wave equation, the heat equation, Laplace’s equation, and Burgers’ equation. These equations are called model equations because they can be used to “model” the behavior of more complicated PDEs. For example, the heat equation can serve as a model equation for other parabolic PDEs such as the boundary-layer equations. All of the present model equations have exact solutions for certain boundary and initial conditions. We can use this knowledge to quickly evaluate and compare numerical methods that we might wish to apply to more complicated PDEs. The various methods discussed in this chapter were selected because they illustrate the basic properties of numerical algorithms. Each of the methods exhibits certain distinctive features that are characteristic of a class of methods. Some of these features may not be desirable, but the method is included anyway for pedagogical reasons. Other very useful methods have been omitted because they are similar to those that are included. Space does not permit a discussion of all possible methods that could be used.
