Consider the dynamic simulation of an object that is sliding across a level surface and experiencing friction. The friction is a lower kinetic friction if the object is sliding faster than some threshold speed, and otherwise a higher static friction.
We can get rid of most of the physical quantities in this system and simply say that the object decelerates at rate a_k if it is faster than some (absolute) speed v_k, and otherwise it decelerates at rate a_s.
We restrict 0 < a_k <= a_s.
We can call the position of our object x, and denote time by t, and then describe its dynamics with a differential equation as follows:
x(t) = t * x'(t)
x'(t) = t * x''(t)
x''(t) = (if |x'(v)| <= v_k then a_s else a_k) * -signum(x'(t))
If we want to simulate this system with the Euler method with a fixed time step, the speed x' will (in general) never become zero, as it will always be incremented or decremented by either a_s or a_k.
What is an explicit Nyström method that can be applied to this system with a fixed time step, such that the speed x' provably converges towards zero in general.
Additionally, it would be nice if it converges at a certain rate O(t^a), with a <= -1 as soon as the simulation reaches the discontinuities in x'''.