Implicit integration for FLIP?

Question

I have problem with volume loss in FLIP simulation.

Unfortunately it's necessary to obey the CFL condition when using explicit integration methods (RK2 in my case) to advance particle positions using the Fluid Implicit Particle method, otherwise fluid volume gets lost over time (simulation becomes unstable).

It turns out that in my case its very expensive to take several sub-iterations per frame and obey the CFL condition.

I thought the problem with volume loss could be solved by using an implicit integration method, that'd make the fluid unconditionally stable and would allow me to take larger time steps without obeying the CFL condition and performing the costly sub-iterations.

Does such implicit formulation of FLIP exist? If not, is there another way how to alleviate (or completely remove) volume loss in FLIP (other than CFL)?

Answer 1

In the computer graphics community there exist many approaches that approximate the advection operator of the Navier-Stokes or Euler equations with unconditional stability by particle tracing that are somehow implicit in time. The basic idea is to use a negative time steps and therefore trace particles backwards in time. At the grid points, where velocity samples are located, the particles are integrated to find the position where the particle originated and use and interpolated velocity for the update. Stam 1 introduced this concept into the computer graphics community, although the concept might have appeared somewhere else before. Zhu and Bridson 2 used FLIP for incompresible flow, but as far as I see use explicit integration for the particle tracing (positive time step). Maybe the methods in 1 and 2 can be combined to get an unconditionally stable FLIP integrator.

However, I don't know if this will lead to an improved volume conservation. Please also note that I made the experience that higher CFL numbers imply a significant reduction in accurary (for the velocity field) and maybe also result in volume loss.

[1] Stam, J. Stable fluids. SIGGRAPH Proceedings, 1999, 121-128

[2] Zhu, Y.; Bridson, R. Animating Sand As a Fluid ACM Trans. Graph., ACM, 2005