I am trying to simulate a rope object. The formulation I understand is an array of particles, connected by springs. These springs have very large k-values, so that the line deforms, but stretches very little. I have concluded that solving this as a function of time is not possible in closed-form because a rope is a generalization of a pendulum (which is not closed-form).
Settling for approximate solutions, then. I need an algorithm that scales well. Examples I have seen use either explicit or implicit Eulerian integration to move the particles. This does not scale.
To see this, consider a rope with n nodes. Apply a large force to one end. Because the rope should not stretch very much, the acceleration at the other end must be immediate.
However, with Eulerian integration, to get any force to the other end requires n steps. I notice an exponential falloff: if the first node accelerates a certain amount, then the adjacent nodes accelerate less (if they accelerate at the same rate, then the algorithm is not stable). Consequently, the nodes adjacent to that node accelerate even slower!
So, for n nodes away, the acceleration is almost negligible. This leads to rope that stretches significantly. If you want to only double the simulation's resolution, you suddenly need to take time steps that are tens or hundreds of times smaller to get similar behavior.
I am looking for a simple method that solves this problem--i.e., higher resolution simulations converge to the solution with only polynomial time extra computation. A full library of matrix and linear algebra techniques is available. My knowledge of classical mechanics is very good, and I know some numerical analysis.