Support cell index

Support Cell Index Definition

Each IB stencil point has a global index \(k\) in the spanwise direction. Before evaluating the interpolation or regularization operators, this global index must be mapped to either a local index k_loc (if the point lives on myid) or a support cell index k_sup (if the point lives on a neighboring rank). This section documents that mapping for both face-based (\(w\)) and center-based (\(u\), \(v\), \(c\)) variables.

The two subroutines that implement this mapping are:

global_to_local_face(k_global, k_sup, k_loc, rank) — for \(w\) (face index)
global_to_local_center(k_global, k_sup, k_loc, rank) — for \(u\), \(v\), \(c\) (center index)

Both return the same three outputs:

Output	Meaning
`rank`	The rank that owns `k_global` (`myid`, `prev`, or `next`)
`k_loc`	Local array index in `F(:,:,k_loc)` — valid only when `rank == myid`
`k_sup`	Support buffer index in `F_supp(:,:,k_sup)` — valid only when `rank /= myid`

The support buffer on each rank has size 2*suppz+1 in the \(z\) direction, storing suppz planes from the left neighbor followed by suppz+1 planes from the right neighbor (or vice versa depending on context).

Support Buffer Layout Summary

For any rank myid, the support buffer F_supp(:,:, 1:2*suppz+1) is organized as follows:

Index in F_supp:   1        ...   suppz+1  |  suppz+2   ...   2*suppz+1
                   ◄── from left neighbor ──┤──── from right neighbor ──►
                   (rank prev / nprocs-1)   │   (rank next / rank 0)

A global index outside the range [1, 2*suppz+1] after the mapping is a fatal error, caught by both subroutines:

if (k_sup < 1 .or. k_sup > 2*suppz+1) stop 'Error: zi_supp out of [1..suppz*2+1]'

Case 1 — Interior Rank for center index / All rank for face index

This case applies to:

All ranks for the face index (global_to_local_face)
Interior ranks (0 < myid < nprocs-1) for the center index (global_to_local_center)

Support cell index – interior rank

For rank \(i\), the neighboring ranks are prev = i-1 and next = i+1. A global index \(k\) belonging to the left neighbor (rank \(i-1\)) maps to:

\[k_{\text{supp}} = k - kg2_{i-1} + \text{supp}_z + 2\]

which places it in the lower portion of the support buffer (indices \(\text{supp}_z - 1\) to \(\text{supp}_z + 1\) as shown in the schematic).

A global index \(k\) belonging to the right neighbor (rank \(i+1\)) maps to:

\[k_{\text{supp}} = k - kg1_{i+1} + \text{supp}_z + 1\]

which places it in the upper portion of the support buffer (indices \(\text{supp}_z + 2\) to \(\text{supp}_z + 3\)).

In code:

! Left neighbor (rank == prev)
k_sup = k_global - kg2_global(prev) + suppz + 2

! Right neighbor (rank == next)
k_sup = k_global - kg1_global(next) + suppz + 1

The local index for a point owned by myid is simply:

k_loc = k_global - kg1_global(myid) + 1
k_sup = -1   ! not used

Periodic Boundary Condition and Center Index

Apart from the general interior rank case, special treatment is needed at the domain boundaries to account for the periodic boundary condition. Since the periodic condition for face-based variables (\(w\)) is handled identically to the ghost cell treatment, only the center index special cases are documented here.

The periodic boundary condition enforces the following equivalences for center-based variables (taking \(u\) as an example):

U(:,:,1) = U(:,:, nzg_global-2)  ! rank 0, local k=1 ← rank nprocs-1, local k=nzg-2
U(:,:,2) = U(:,:, nzg_global-1)  ! rank 0, local k=2 ← rank nprocs-1, local k=nzg-1
U(:,:,3) = U(:,:, nzg_global  )  ! rank 0, local k=3 ← rank nprocs-1, local k=nzg

This means that local index k=2 on rank 0 holds the same physical value as local index k=nzg-1 on rank nprocs-1. As a result:

Rank 0 (right neighbor to rank nprocs-1): when rank nprocs-1 looks up data from rank 0, the effective index must be shifted one position to the left relative to the interior formula, because the first meaningful plane of rank 0 starts one index later than a standard interior rank would.
Rank nprocs-1 (left neighbor to rank 0): when rank 0 looks up data from rank nprocs-1, the effective index must be shifted one position to the right, because k=nzg-1 at rank nprocs-1 is physically equivalent to k=2 at rank 0.

!!! note "Special treatment for k=2 and k=nzg_global-1" The planes at global index k=2 (owned by rank 0) and k=nzg_global-1 (owned by rank nprocs-1) require special handling because they are periodic images of each other:

Regularization: the contribution to these planes is computed as if the point is interior to rank nprocs-1. The result is then transferred to rank 0 and accumulated there. After the solve, the periodic boundary condition propagates the correct value back to rank nprocs-1, ensuring consistency.
Interpolation: since the velocity field already satisfies the periodic boundary condition before the operator is applied, rank nprocs-1 can read data directly from index 3+ in rank 0 without any additional correction.

Case 2 — Right Boundary Rank Accessing Rank 0 (Periodic, Center Index Only)

This case applies to rank nprocs-1 when it needs support cell data from rank 0 (its right neighbor under the periodic boundary condition).

Support cell index – right boundary

In the interior formula, the support index for a right neighbor is:

\[k_{\text{supp}} = k - kg1_{i+1} + \text{supp}_z + 1\]

However, because k=2 on rank 0 is the periodic image of k=nzg-1 on rank nprocs-1, rank 0's effective interior starts one plane later than a standard interior rank. The support index must therefore be shifted one position to the left:

\[k_{\text{supp}} = k - kg1_{1} + \text{supp}_z\]

In code:

! Right neighbor is rank 0 — apply periodic correction
k_sup = k_global - kg1_global(next) + suppz + 1
if (rank == 0) k_sup = k_sup - 1

Case 3 — Left Boundary Rank Accessing Rank `nprocs-1` (Periodic, Center Index Only)

This case applies to rank 0 when it needs support cell data from rank nprocs-1 (its left neighbor under the periodic boundary condition).

Support cell index – left boundary

In the interior formula, the support index for a left neighbor is:

\[k_{\text{supp}} = k - kg2_{i-1} + \text{supp}_z + 2\]

However, because k=nzg-1 on rank nprocs-1 is the periodic image of k=2 on rank 0, the last meaningful plane of rank nprocs-1 extends one index further than a standard interior rank. The support index must therefore be shifted one position to the right:

\[k_{\text{supp}} = k - kg2_{n-1} + \text{supp}_z + 3\]

In code:

! Left neighbor is rank nprocs-1 — apply periodic correction
k_sup = k_global - kg2_global(prev) + suppz + 2
if (rank == nprocs-1) k_sup = k_sup + 1

Support cell index