Kevin T. Chu

SAMR Phase Field Model Development Notes

Fixed "grid anisotropy" bug. It turns out that it was a bug using too small a number of ghost cells when running the code in parallel.
Changed algorithm for computing grad(phi) when computing theta. I now take the average value of grad(phi) computed using central differencing at the centers of the two grid cells immediately next to the face on which the flux is being computed.
I used second-order Runge-Kutta for time integration because it yields an error that is O(dx^2) in space and O(dt^2=dx^4) in time.

Testing showed that near the interface, the gradient is very sensitive to the numerical scheme used to compute it. I suspect that higher-order schemes may help make the gradient calculation less sensitive to the numerical scheme, but I have not tested this yet.
Both 5pt and 9pt stencils now give good results.
The problem with the 5pt scheme in my earlier runs was that the time step was set too large for the heat equation part of the simulation. It lead to inaccuracies in the temperature field near the interface which degraded the evolution of the interface. I have fixed this problem.
The results shown are mostly for uniform grids, but there are two that show the results for a SAMR grid with three refinement levels.

The time to run each uniform grid simulation was run from t=0 to t=0.5 and took about 2.5 minutes using two processors on a MacBook Pro with Intel Core 2 Duo processor.
The time to run each SAMR simulation was run from t=0 to t=0.3 with three refinement levels (ratio to coarser levels = 4, 2) and took about 100 minutes (not dedicated) using two processors on a MacBook Pro with Intel Core 2 Duo processor. The simulation starts out very fast and gets slower as time progresses because the region covered by the fine grid grows larger and larger. This is where higher-order discretizations would help because then we don't need as fine a grid at the finest levels.

Changed algorithm for computing theta. I was using a way to compute the gradient of phi that lead bad estimates for theta.
Fixed bugs in fortran routines for flux.