Euler’s Method

1.5.1 Euler’s Method

The approximate solution of differential equations is a very important task in scientific computing. Nearly all the areas of science and technology contain mathematical models which lead to systems of ordinary (or partial) differential equations. For the step by step simulation of such a system a mathematical model is first set up; i.e., state variables are set up which describe the essential features of the state of the system. Then the laws are formulated, which govern the rate of change of the state variables, and other mathematical relations between these variables. Finally, these equations are programmed for a computer to calculate approximately, step by step, the development in time of the system.

The reliability of the results depends primarily on the quality of the mathematical model and on the size of the time step. The choice of the time step is partly a question of economics. Small time steps may give you good accuracy, but also long computing time. More accurate numerical methods are often a good alternative to the use of small time steps.

The construction of a mathematical model is not trivial. Knowledge of numerical methods and programming helps in that phase of the job, but more important is a good understanding of the fundamental processes in the system, and that is beyond the scope of this text. It is, however, important to realize that if the mathematical model is bad, no sophisticated numerical techniques or powerful computers can stop the results from being unreliable, or even harmful.

A mathematical model can be studied by analytic or computational techniques. Ana- lytic methods do not belong to this text. We want, though, to emphasize that the comparison of results obtained by applying analytic methods, in the special cases when they can be applied, can be very useful when numerical methods and computer programs are tested. We shall now illustrate these general comments using a particular example.

An initial value problem for an ordinary differential equation is to find y(t) such that

The differential equation gives, at each point (t, y), the direction of the tangent to the solution curve which passes through the point in question. The direction of the tangent changes continuously from point to point, but the simplest approximation (which was proposed as

early as the eighteenth century by Euler 17 ) is that one studies the solution for only certain

17 Leonhard Euler (1707–1783), incredibly prolific Swiss mathematician. He gave fundamental contributions to many branches of mathematics and to the mechanics of rigid and deformable bodies, as well as to fluid mechanics.

56Chapter 1. Principles of Numerical Calculations values of t = t n = nh, n = 0, 1, 2, . . . , (h is called the “time step” or “step length”) and

assumes that dy/dt is constant between the points. In this way the solution is approximated by a polygon (Figure 1.5.1) which joins the points (t n ,y n ) , n = 0, 1, 2, . . . , where

Thus we have the simple difference equation known as Euler’s method: y 0 = c,

y n +1 =y n + hf (t n ,y n ), n = 0, 1, 2, . . . . (1.5.2) During the computation, each y n occurs first on the left-hand side, then recurs later on the

right-hand side of an equation. (One could also call (1.5.2) an iteration formula, but one usually reserves the word “iteration” for the special case where a recursion formula is used solely as a means of calculating a limiting value.)

Figure 1.5.1. Approximate solution of the differential equation dy/dt = y, y 0 =

0.25, by Euler’s method with h = 0.5.