periodic_bc

Family: finite_difference
Grid family: cartesian
Kind: boundary_condition
Accuracy: exact (index rewrite)
Applies to: index(u, i − offset) on a periodic axis
Rule file: discretizations/finite_difference/periodic_bc.json
Tags: #finite-difference #boundary-condition #periodic #index-rewrite #P0

Rewrite

periodic_bc is not a discretization scheme — it is one of the downstream boundary-condition rewrite rules that consume the domain’s boundary_conditions block (esm-spec §11.5) and rewrite a lowering’s concrete index expressions at the boundary cells. A scheme rule (e.g. centered_2nd_uniform or weno5_advection) emits the interior closed form with bare index(u, i ± k) accesses; this rule fires afterward on every such access whose axis the domain marks periodic, wrapping the index back into range.

The rule matches an index access whose index expression is a metavariable shift $i − $offset and replaces it with the modulo-wrapped form:

applies_to:  index($u, $i − $offset, $rest)
where:       var_has_grid($u, $g)          # $u lives on grid $g
             dim_is_periodic(x, $g)         # axis x of $g is periodic
replacement: index($u, mod(($i − $offset) + Nx, Nx), $rest)
region:      $side

$rest carries any trailing indices unchanged (so the rule is axis-local on a multi-dimensional array), and region: $side scopes the rewrite to the boundary side that triggered it. Nx is the periodic axis length supplied by the grid.

After substitution, a stencil neighbor that would read off the low or high end of the axis instead reads the wrapped-around interior cell:

$$\texttt{index}(u,\; i - \text{off}) \;\longrightarrow\; \texttt{index}\!\left(u,\; (i - \text{off} + N_x) \bmod N_x\right).$$

The + Nx before the mod keeps the argument non-negative so the wrap is correct for left-edge accesses (i − off < 0) under truncating-modulo semantics.

Guards

Two where guards gate the rewrite so it fires only where periodicity actually applies:

Guard	Binds / checks
`var_has_grid`	binds `$g` to the grid the accessed variable `$u` lives on
`dim_is_periodic`	fires only when axis `x` of grid `$g` is declared periodic in the domain’s `boundary_conditions`

Because the guards read the (grid, axis-periodicity) pair from the domain, the same closed scheme lowering composes with a periodic axis here and with a Dirichlet/Neumann fill row elsewhere — the scheme rule itself stays BC-agnostic. This is the mechanism referenced by the BC handoff tables on the scheme pages (centered_2nd_uniform, weno5_advection).

Convergence

This is an index-rewrite rule, not a manufactured-solution scheme, so it has no standalone empirical order of accuracy — periodic wrapping is exact (it relabels indices; it introduces no truncation error). Its acceptance signature is therefore a rewrite fixture rather than an MMS convergence sweep: the Layer-B convergence fixture under discretizations/finite_difference/periodic_bc/fixtures/convergence/ carries applicable: false (with a tracked skip_reason), and the authoritative check is the AST-rewrite golden under fixtures/rewrite/, which pins index(T, i − 1, j) on an x-periodic grid rewriting to the mod-wrapped form. The correctness of the scheme that this rule supports is verified on a periodic domain by that scheme’s own convergence fixture (e.g. the centered_2nd_uniform sweep on sin(2πx)).