[CIG-CS] Particle problems and Semi-Lagrangian Schemes for Gamr

[Walter Landry]

Hello Everyone,

Recently, I did some experiments with Q2-Q1 elements and particles
with Gale.  Because of the higher order Q2 elements, it takes
literally hundreds of particles to get a solution accurate to a part
in 10^3 in 2D.  The deal.II finite element AMR code uses Q2 elements,
so it would run into the same problem with those kinds of particles.

One solution is to represent the properties as a field and
use semi-lagrangian methods.  It has been used extensively for all
sorts of advection problems, though it can be somewhat diffusive.
However, I have seen some pretty impressive results with high order
fields (e.g. 4th order).  In any case, I think a minor amount of
diffusion would be OK since this is for advecting material properties, not
material interfaces.

However, in my brief research, I could not find anyone who has used it
for a realistic Stokes flow problem.  Almost everyone (Gale, DOUAR,
I3ELVIS, SLIM3D, LaMEM, Milamin) uses particles.  So my main
reservation would be that no one has used it for this kind of problem.
So I do not know if there are any problems that will somehow make it
infeasible.

With that said, even with a finite difference code I am inclined to
implement semi-lagrangian methods before particles, since it seems
like a cleaner, simpler method.

Cheers,
Walter Landry
walter at geodynamics.org