Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 4

IMPLICIT EULER METHOD

Some differential equations, such as stiff equations, are difficult to approximate using explicit methods because the stability condition imposes severe restrictions on the time step. It is therefore very useful to introduce a new class of methods, called implicit methods. In this chapter we explain some basic properties of stiff equations by analyzing a simple model equation. Then we introduce the implicit Euler method, which is the most basic implicit method. At the end of the chapter a very simple variable-step-size strategy is explained. This strategy shows the fundamental idea behind more complex strategies used in higher-order methods.

4.1 Stiff equations

Many applications lead to Stiff differential equations. A simple example of a stiff equation is

which can also be written as

(4.1)

This equation can be solved exactly using lemma 1.2. The exact solution is

(4.2)

To gain insight into the behavior of the solution, instead of analyzing the exact solution (4.2), we prefer to use a general procedure to obtain the dominant terms of this solution. This procedure has the advantage that it can be applied to more general problems where the exact solutions ...