How does the time needed for force propagation increase with the discretization (using symplectic Euler schemes)?

I am trying to model a physical system by (lets say the system is a long deformable object, on which I can apply forces). It can be described by Cosserat Rod theory and discretized, by modelling it trough several spheres. If I now apply forces on one of the spheres, to me it seems, like the force reaches the other spheres far slower if I have many spheres. I am solving the positions based on a symplectic Euler scheme and using a constraint to keep the spheres at approximately the same distance from each other in order to account for deformation of my object. Through this constraints and the Euler scheme, the force gets transported. I am now wondering, if there is a theoretical expectation, how the time the force propagation takes, will evolve with the number of spheres for example. An idea, what theory could be used to describe this, would be very helpful, thanks!

• Is your scheme explicit or implicit in time? Commented Sep 25, 2023 at 23:20
• Thanks for the response! It is an explicit scheme in time. Commented Sep 26, 2023 at 9:57
• Do you also reduce the time step when the "space disretization" gets denser? I do not know your particulars, but suspect that the force propagation is limited to one sphere distance per integration step. This becomes unphysical if the integration step is too large. Commented Sep 26, 2023 at 11:35
• @LutzLehmann I see. I do not change the time step with changing the space discretization. The force propagation takes place over the constraint, that tries to enforce a constant length between the spheres. This enforcing will go from one side of the object to the other side of the object, trying to enforce the constraint between two adjacent spheres. This enforcing of constraint is repeated several times before outputting the positions of the spheres of the next frame. Is there a (theory for) a direct dependence between the denseness of space discretization and time step? I would be grateful! Commented Sep 26, 2023 at 16:12
• In an explicit scheme, indeed in each time step information is only propagated from one degree of freedom to those it is explicitly coupled with. In your case, if each sphere is only coupled to its geometric neighbors, then in each time step you only propagate forces from one sphere to the next, not along the entire chain. Commented Sep 26, 2023 at 17:09