Stable Simulation: Avoiding Overshoot

CS 480 Lecture, Dr. Lawlor

Almost all interesting simulations involve some sort of closed-loop feedback, where a change in A causes a change in B which in turn affects A again.

For example, consider the velocity of an object including drag forces from air resistance.
    F = -k * V
    A = F/m  (from F = mA)
    dV / dt = A

Here F is the net force on the object, V is the object's velocity, k is the object's wind resistance (newtons of resistance per meter per second of velocity), A is the object's acceleration, and m is the mass.

If drag is the only force involved, and discretizing to first order, we get:
    dV = (-k*dt/m) * V
    V = V + (-k*dt/m) * V = V + speed * V
where we've defined speed = (-k*dt/m).

That is, the change in V itself varies with V.  If that speed term (-k*dt/m) is positive, then V will feed back on itself leading to exponential growth.  Luckily, speed is always negative because real drag constants, timesteps, and masses are all positive.  In the theoretical non-discretized equations, a negative speed term causes V to drop exponentially, though possibly with a managably small exponent.  A small negative speed term causes the discrete V to also drop exponentially.

But weird stuff starts happening if the speed term gets negative enough:
Anytime speed<-2.0, V is going to spiral out of control fairly quickly.  This is similar to an inexperienced driver skidding on ice: they steer too hard, overcorrecting the car's orientation, and end up making the skid worse instead of better.  The effect is that the simulation "blows up", where the points have vanished off the screen, or hit floating-point infinity or not-a-number.  Not good.

Note that speed = -k*dt/m, so speed is going to get out of control if:
Many simulations have a stability limitation on the timestep.  If your steps get bigger than this, you can expect the simulation to explode.