flux_1d_ppm

Stencil

Continuous form

flux_1d_ppm discretizes the conservative 1D advective flux on a uniform Cartesian grid,

using the piecewise-parabolic method of Colella & Woodward (1984): each cell carries a sub-grid parabolic profile reconstructed from cell-averaged data, the §4 monotonicity limiter clips the parabola when an extremum sits on the wrong side of the cell, and the face flux is the area-averaged value of the upwind cell’s profile over the swept volume |c|·dx of one time step:

Upwind cell selection is encoded directly in the AST via ifelse(c >= 0, int_left, int_right) — no caller-side branching on the sign of the Courant number.

Discrete operator

The closed-form composition is recorded in the JSON rule’s edge_value_stencil, limiter, flux_integral, and flux_form blocks. At a high level:

1. 4th-order edge interpolation (CW84 eq. 1.6) — same coefficients as ppm_reconstruction’s q_right_edge stencil:

   $$q_{i+1/2}\;=\;\tfrac{7}{12}(q_{i-1}+q_i)\;-\;\tfrac{1}{12}(q_{i-2}+q_{i+1}).$$

   The same formula shifted by ±1 cell yields the left and right edges of each upwind cell’s parabola (ql_L_raw, qr_L_raw, ql_R_raw, qr_R_raw).

2. Colella & Woodward (1984) §4 monotonicity limiter — applied independently to each upwind cell’s reconstruction. The is_extremum case flattens to a constant when q_i is a local extremum within the stencil; the overshoot_left / overshoot_right cases pull the offending edge inward when the parabola’s interior extremum sits past the opposite edge. Encoded as a closed-form ifelse AST identical to the vertical_remap limiter block and bit-for-bit equivalent to the former _ppm_limit_cw84_sym (retired from src/operators/reconstruction.jl in the operator-porting campaign).

3. Courant-fraction flux integral — closed-form parabolic antiderivative evaluated over the swept volume of the upwind cell. With dq = qr - ql and q_6 = 6·(q_i − (ql + qr)/2),

   $$\text{int\_left}\;=\;qr - \tfrac{|c|}{2}\!\left[dq - q_6\!\left(1 - \tfrac{2}{3}|c|\right)\right],$$ $$\text{int\_right}\;=\;ql + \tfrac{|c|}{2}\!\left[dq + q_6\!\left(1 - \tfrac{2}{3}|c|\right)\right].$$

4. Upwind selection + face velocity —

   $$F_{i+1/2}\;=\;v\cdot\text{ifelse}(c \ge 0,\;\text{int\_left},\;\text{int\_right}).$$

The full machine-readable AST lives in the JSON rule file.

Bindings

The face-staggered $c and $v follow the same per-face binding contract as lax_friedrichs_flux — the analog of the slope-ratio $r in the limiter rules. The 6-point stencil reaches three cells past the right cell of the interface; the former imperative reference flux_1d_ppm_arrayop (retired from src/operators/flux_1d.jl in the operator-porting campaign) resolved this with a ghost-extended q_ext. ESS does not yet declare how a rule consumes ghost cells — see the rule’s schema_gaps block.

Composition

flux_1d_ppm is the high-order PPM sibling of lax_friedrichs_flux: same op = flux shape and per-face binding contract, but composes ppm_reconstruction edge values, the Colella-Woodward §4 monotonicity limiter (same closed-form AST as the vertical_remap limiter block), and a Courant-fraction flux integral, with ifelse upwind selection in place of the LF abs-based first-order upwind. The cell-centred tendency follows from the FV divergence $T_i = -(F_{i+1/2}\cdot\Delta x_{i+1/2} - F_{i-1/2}\cdot\Delta x_{i-1/2})/A_i$ (or composes with divergence_arakawa_c on a C-grid).

Convergence

Numeric coverage today lives in the rule’s own test test/test_flux_1d_ppm_rule.jl, which exercises the closed-form face flux (positive- and negative-Courant cases, CW84 limiter inactive and active). The former imperative flux_1d_ppm! / flux_1d_ppm_arrayop operators and their cubed-sphere transport tests were retired (operator-porting campaign and the cubed-sphere grid retirement).

The hand-pinned single-face fixture at fixtures/canonical/ documents the closed-form face flux on a smooth sinusoidal profile (positive- and negative-Courant cases; CW84 limiter inactive); the same values are exercised inline in test/test_flux_1d_ppm_rule.jl against the closed-form CW84 limiter AST + the closed-form Courant integral.

Reference

• Former imperative reference: flux_1d_ppm! and flux_1d_ppm_arrayop in src/operators/flux_1d.jl; symbolic-tracing-safe limiter _ppm_limit_cw84_sym in src/operators/reconstruction.jl. Both files were retired in the operator-porting campaign (commit dce15e6).
• Theory: Colella & Woodward (1984), JCP 54(1):174-201, eqs. (1.5)-(1.10) for the parabolic reconstruction and the (1.7)-(1.10) monotonicity limiter; the §4 Courant-fraction flux integral is the standard PPM upwind flux. Flux-form 1D advection composition follows Lin & Rood (1996) MWR.
• First-order sibling: lax_friedrichs_flux (F_{i+1/2} = max(c,0)·q_i + min(c,0)·q_{i+1}).
• Shares its edge interpolation with ppm_reconstruction and its monotonicity limiter with vertical_remap.