[Physics] Why is Verlet integration better than Euler integration?

Question

Hello! Can someone explain me why Verlet integration is better than Euler integration? And why RK4 better than Verlet? I don't understand why it is a better method :/

Answer 1

If everything just coasts along in a linear way, it wouldn't matter what method you used, but when something interesting (i.e. non-linear) happens, you need to look more carefully, either by considering the non-linearity directly (verlet) or by taking smaller timesteps (rk4).

Answer 2

The Euler method is a first order integration scheme, i.e. the total error is proportional to the step size. However, it can be numerically unstable, in other words, the accumulated error can overwhelm the calculation giving you nonsense. Please note, this instability can occur regardless of how small you make the step size or whether the system is linear or not. I am not familiar with verlet integration, so I can not speak to its efficacy. But, the Runge-Kutta methods differ from the Euler method in more than just step size.

In essence, they are based on a better way of numerically approximating the derivative. The precise details escape me at the moment. In general, the fourth order Runge-Kutta method is considered the workhorse of the integration schemes, but it does have some disadvantages. It is slightly dissipative, i.e. a small first derivative dependent term is added to your calculation which resembles an added friction. Also, it has a fixed step size which can result can make it difficult to achieve the accuracy you desire. Alternatively, you can use an adaptive stepsize scheme, like the Runge-Kutta-Fehlberg method, which gives fifth order accuracy for an additional 6 function evaluations. This can greatly reduce the time necessary to perform your calculation while improving accuracy, as shown here.

Answer 3

The Verlet method is is good at simulating systems with energy conservation, and the reason is that it is symplectic. In order to understand this statement you have to describe a time step in your simulation as a function, f, that maps the state space into itself. In other words each timestep can be written on the following form.

(x(t+dt), v(t+dt)) = f(x(t),v(t))

The time step function, f, of the Verlet method has the special property that it conserves state-space volume. We can write this in mathematical terms. If you have a set A of states in the state space, then you can define f(A) by

f(A) = {f(x)| for x in A}

Now let us assume that the sets A and f(A) are smooth and nice so we can define their volume. Then a symplectic map, f, will always fulfill that the volume of f(A) is the same as the volume of A. (and this will be fulfilled for all nice and smooth choices of A). This is fulfilled by the time step function of the Verlet method, and therefore the Verlet method is a symplectic method.

Now the final question is. Why is a symplectic method good for simulating systems with energy conservation, but I am afraid that you will have to read a book to understand this.