Understanding the Limitations of the Euler Method in Computational Physics

[spaghetti3451]

This is an extract from my third year notes on 'Computational Physics':

The Euler method is inaccurate because it uses the gradient evaluated at the initial point to
calculate the next point. This only gives a good estimate if the function is linear since the truncation error is quadratic in the step size.

My question is this:

If the function is linear, then the Euler method must give the exact answer as the gradient lies on the line. So, why does it say that the Euler method only gives a good estimate if the function is linear.

Any ideas? Is it wrong?

Should it be the Euler method only gives a good estimate if the function is approximately linear, so that the quadratic and higher order terms of the function in that case are much much smaller than the linear term so that the error is minimal?

 

[Filip Larsen]

Perhaps you are mistaking a linear field (the linear function your note mentions) for a solution linear in time? For instance, a harmonic oscillator may be described by a linear field, but since the solutions are circular (in state space) Euler's method will introduce significant truncation error.