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Recently I have stumbled upon this video of M. J. Gander https://www.youtube.com/watch?v=dn5vqN8ezuE and the coresponding notes that he wrote on Time Parallel Time Integration and I find it a quite interesting topic.

I would like to ask about the possible gain of applying the parareal algorithm to particle simulations (e.g. the SPH method used in fluid dynamics).

I have a specific code on which I would like to try it but I would like to know first if it will even work.

The framework in which I would like to apply it is as follows: Say that the problem is governed by an advection-diffusion equation. For reasons that are a bit involved, this equation is solved by splitting the problem into two sub-problems: an advection and a diffusion problem. These are solved sequentially. Suppose that both sub-problems can be solved using particles (i.e. no finite differences, finite volumes etc discretization).

Could parareal be applied to this problem given the fact that I use the splitting? Is it possible to apply parareal only to one of the sub-problems (possibly only the advection part)?

Any feedback is welcome! Feel free to ask for any other clarification.

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Parareal is fairly effective for parabolic problems, but doesn't work well for hyperbolic problems. So it wouldn't be effective for the advection part, but could be effective for the diffusion part. Making it work well for hyperbolic problems is an area of ongoing research. Note that the parallel efficiency of parareal is quite poor, so if you haven't yet parallelized in space (i.e. over the particles) then you'll get a much bigger benefit from that.

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  • $\begingroup$ Thanks for the answer! About the comment on hyperbolic problems, I was wondering since the advection equation is solved also using particles (i.e. not discretizing the advection term) then what I have to do is only integrate with an Euler method or Runge Kutta method in order to move the particles (like it would happen if a semi-lagrangian method would be used for the full equation). Having this in mind, it is not very clear to me why parareal would not work for this case. As for the parallelization in space, yes they do exist in the code I use. $\endgroup$ – Riri Feb 12 at 9:10
  • $\begingroup$ I believe the ODEs that you need to solve for the particles have the same spectral character (i.e., eigenvalues on or near the imaginary axis) as hyperbolic PDE semi-discretizations. That's why I believe parareal will also be inefficient for them. $\endgroup$ – David Ketcheson Feb 12 at 10:59
  • $\begingroup$ I think the only real reason to use time parallelism (with its much lower parallel efficiency) is if you have hit the strong scaling limit for parallelism in space. $\endgroup$ – David Ketcheson Feb 12 at 11:00
  • $\begingroup$ I will work my way a bit around that and maybe come back with another question. thank you! $\endgroup$ – Riri Feb 12 at 13:12

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