The Euler-Cromer method

Although the Euler method is simple to apply, it is not very accurate. Here we discuss one improvement to the Euler method that is particularly useful for investigating oscillating systems. As an example of such a system consider a harmonic oscillator. This could be a mass attached to an ideal spring or a simple pendulum of small amplitude.

For a pendulum of length l, and mass m, the angle with respect to the vertical satisfies

For small amplitude, we can use the approximation

(where  is measured in radians) so that

This is the equation for simple harmonic motion with angular frequency

For such a system the total energy

is conserved.

For a numerical method to give a physically correct result the total energy should be conserved or at least it should remain approximately constant over many periods of oscillation. Let’s see if the Euler method does this. To reduce the number of symbols, use  as the unit of time and mgl as the unit of energy. The equations become

and

  To apply Euler’s method, use the angular velocity

as a second dependent variable. The basic Euler step is

 To see if the energy is conserved consider

Hence we conclude that the Euler method has the undesirable property that it makes E_n increase with time.

A simple modification to the Euler method goes some way to fixing this problem. Replace the basic Euler step by

This is called the Euler-Cromer method

The reason why this modification works is not trivial but for those of you who are interested in finding out why, read on!

With this change you can show that

Now the term proportional to  can be positive or negative. For simple harmonic motion the exact solution is

so that

This means that the average value of  over one complete period of oscillation is zero. A more complete analysis shows that for oscillatory motion the energy increase over one complete cycle using the Euler-Cromer method is proportional to  whereas for the Euler method the energy increase is proportional to .