Parallelization

The implementation uses MPI to parallelize the most expensive parts of the pipeline.

Spreading vorticity to the grid

• Each rank handles a strided subset of grid points (round-robin across the flattened (k,j,i) index ordering).
• For each grid point, it collects contributions from nearby elements via the chosen kernel and periodic tiles.
• Local contributions are accumulated in a rank-local buffer; a final MPI.Allreduce (SUM) builds the global grid vorticity.

Benefits: embarrassingly parallel, no halo exchange, and no custom domain decomposition needed.

Poisson solve

• The FFT-based Poisson solver currently runs on rank 0, computes the full solution (Ux,Uy,Uz), and broadcasts to all ranks.
• For very large problems, Poisson provides interfaces for iterative or multigrid solvers. A hybrid solver can try FFT first and fall back if the residual is above a threshold.

Interpolation to nodes

• Each rank handles a strided subset of nodes and accumulates interpolated velocities from nearby grid cells using the same kernel family.
• Per-rank results are combined with MPI.Allreduce.

Time stepping and adaptivity

• Node velocities are computed in parallel, and RK2 updates are carried out locally with final periodic wraps per step.
• Adaptive dt is computed from the global maximum grid speed via MPI.Allreduce (MAX).

Checkpointing and I/O

• Checkpoints (CSV, MAT, JLD2) are written on rank 0 to avoid file contention. Rank 0 aggregates necessary data and saves.

Figures from the thesis

Recommended figures:

• Schematic of the Lagrangian–Eulerian pipeline.
• Strong/weak scaling curves (if available).

Place images extracted from the thesis into docs/src/assets/ and reference here.