I'm implementing a mass-spring solid mechanics solver and I'd like to use the Velocity-Verlet time integration scheme. However, I cannot find anything about the maximum stable time step -- either saying there is none and it's unconditionally stable like an implicit scheme or that there is one and it's bound by something like the explicit schemes.

I would think the maximum time step is related to the maximum force and the minimum distance between the particles if there is one, but I can't seem to find anything. What is the stable time step limit for the Velocity-Verlet integration?


1 Answer 1


In the following, I am basically rephrasing p. 24 of "MCMC using Hamiltonian Dynamics" by Radford Neal.

At least for leapfrog integration, one can analytically calculate the maximum time step for quadratic Hamiltonians. For such systems, a leapfrog step is a linear mapping $(q(t), p(t) \mapsto (q(t+\epsilon), p(t+\epsilon))$. Stability then depends on the magnitude of the eigenvalues of the matrix representing this mapping. For a harmonic oscillator with force constant $k$, these eigenvalues are given by $(1 - \frac{k\epsilon^2}{2}) \pm \epsilon \sqrt k \sqrt{\frac{k \epsilon^2}{4} - 1}$. So for $\sqrt{k}{\epsilon} < 2$, the eigenvalues are complex and have a squared magnitude of $1$. The system simulated with this timestep will therefor be stable. On the other hand, if $\sqrt{k} \epsilon > 2$, the eigenvalues are real and at least one of them will have an absolute value $> 0$. This means a simulation with this time step will be unstable when you iterate this scheme.

Now this is for leapfrog integration, but as Velocity Verlet integration is closely related, I think one can calculate a similar stability criterion and if not, then I think the result for leapfrog integration should also be comparable to the maximum time step for Velocity Verlet integration. Specifically for your problem, I guess the maximum time step will be given by the spring constant of the stiffest spring; "mass-spring solid mechanics" sounds suspiciously like harmonic potentials to me.


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