Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 2

METHOD OF EULER

Essentially all concepts introduced in Chapter 1 have their counterpart in this chapter. We introduce the simplest difference method that one can use to approximate an initial value problem. Most of the fundamental concepts needed to understand numerical approximations to differential equations follow the concepts introduced here for the method of Euler. These concepts are more complicated; therefore, one should understand the procedures for differential equations before one starts with the difference approximations. New concepts are accuracy, truncation error, and stability region. Also, a discrete version of Duhamel’s principle is discussed. General one-step methods are defined and a section is devoted to tests of correctness of a program.

2.1 Explicit Euler method

There are many methods of computing approximations to the solution of an initial value problem. The explicit Euler method,¹ which we discuss here, is the most easily understood and implemented. To describe it, let us first introduce the concepts of grid and grid function.

Starting form the initial time in our problem, say t = 0, we divide the t-axis into subintervals of length k > 0 and obtain a grid (see Figure 2.1). The endpoints t_n = nk, n = 0, 1, 2, … of the subintervals are called grid points. A grid function g_n = g(nk) is a function that is denned on the grid. The explicit Euler method for an initial value problem

Figure 2.1 Grid starting at t = 0.

will be derived below, but we start ...