Smoothing of absolute value and sign functions for numerical integration

Question

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ps.

At some point the equations become so stiff that s becomes negative and if I improve tolerances of the numerical integrator, it crashes. To solve this problem, I'd like to approximate the ODEs with s_new=abs(s) and ps_new=-ps around the critical region (for example if(s<1.0e-8)). The problem is that these are not smooth. Numerical integrators (I use BDF) break down.

One solution that I came across is to use arctan for abs and erf for sign.

It was hard for me to find anything in literature on it. If you know of a better approach, please let me know. Note, there is conservation of energy, so there exist f(s,ps) such that df/dt=0

Answer 1

You can use: $2\Phi(x/\sigma) - 1$ as a good approximation to the signum function, where $\sigma$ is small relative to $x$.

Answer 2

Are you sure that the failure of the integrator is due to lack of smoothness? It is quite possible to integrate over non-smooth conditions with standard techniques, depending on the character of the system.

If you say that you have a stiff problem, I would imagine that that would be the cause more than anything. Have you tried using a multi-step method instead of a BDF? An Adams-Moulton implicit linear multistep method might be more suitable, and you might be able to avoid this problem altogether. It would help if you posted your system in detail.