Fundamental Problems

As you saw from the derivation's step (5), the two criteria needed to make the Verlet algorithm exact are constant acceleration and constant time step. For most practical cases we cannot guarantee either of these criteria. Of course, there are some simple cases where both criteria will be met. For example, simple projectile physics simulated by a physics engine which uses fixed time steps will yield perfect results. As soon as you add friction, springs or constraints of any kind you nullify the constant acceleration criterion, and adapting your time step to your game's framerate will nullify the constant time step criterion.

I am not going to address the constant acceleration criterion, mainly because explicit integrators (such as this one) must assume the constant acceleration principle, which is violated the instant you start simulating a complex system. Take the example of a point mass connected to a spring: as soon as it starts to move, the spring force, and thus the acceleration, changes. Even equation (1) required that the acceleration be constant throughout a time step. The error introduced by assuming that a_{i-1 =} a_i is actually implicit in our choosing an explicit scheme without knowing how the acceleration changes. We are effectively setting d(a)/dt (a.k.a. the "jerk") to 0.0. The other reason I will ignore this issue is that, empirically, the standard Verlet method already handles changing accelerations better than the Euler method (or even the improved variation on the Euler method), as long as the time step is fixed. Note that the Time-Corrected Verlet will be identical to the original Verlet when the time step is fixed. Observe the following graphs:

The error introduced by the constant time step assumption is something that can be easily ameliorated. This will be shown in the section entitled "A Simple Time-Correction Scheme".

Implementation Problems

A large source of inaccuracy when using the Verlet scheme stems from the improper specification of initial conditions. Looking at equation (6) and trying to fit it into the form of equation (1) (which may be more familiar) may yield (improper) reasoning like this: the first (x_i) term is the position contribution, the second (x_i - x_i-1) term is basically the velocity contribution, and the (a * dt * dt) term is clearly the acceleration contribution. So when simulating the traditional projectile path, setting x₀ = 0.0, and x_-1=x₀ - v₀ * dt₀, and running the simulation from there will give the wrong results, as can be seen in the following graph:

This is because both the second and third terms include acceleration information. So, when setting the current state explicitly (i.e. the position and velocity initial conditions), remember to use equation (2). Shooting simulations depend upon starting with accurate initial conditions. The larger the initial time step, the less accurate your computed initial state will be if you do not use equation (2).

A Simple Time-Correction Scheme