weno5_advection

Family: finite_volume
Grid family: cartesian
Kind: scheme
Accuracy: O(dx⁵)
Applies to: div(U·q), dim=x
Rule file: discretizations/finite_volume/weno5_advection.json
Tags: #finite-volume #weno #jiang-shu #advection #high-order

Lowering

The rule is the canonical exemplar of an ESD nonlinear scheme that lowers a §4.2 PDE operator to a closed arrayop expression in the §4.2 op vocabulary — the sibling of centered_2nd_uniform for linear schemes. There are no scheme-specific coefficient blobs and no off-spec match keys: applies_to matches against div (a §4.2 op) applied to the advective flux *($U, $q), and the lowering is a single arrayop whose body combines index, +, -, *, /, ^, and ifelse. ESS bindings (any binding) can evaluate the lowering by walking the AST through the existing arrayop / broadcast evaluator — no new kernels.

applies_to:  div(*($U, $q), dim=x)
replacement: arrayop(
               output_idx = [$x],
               expr       = (F_E − F_W) / dx,
               args       = [$U, $q]
             )

The face flux F = U_face · q^upwind is built from a cell-to-face average of U and a Jiang–Shu (1996) WENO5 reconstruction of q. Upwinding is encoded directly in the AST as

$$F_{i+1/2} \;=\; U_{i+1/2}\;\cdot\; \mathrm{ifelse}\!\left(U_{i+1/2} > 0,\; q^{L}_{i+1/2},\; q^{R}_{i+1/2}\right),$$

with the cell-to-face velocity average $U_{i+1/2} = \tfrac12(U_i + U_{i+1})$ and a symmetric expression at the west face. The right-biased reconstruction $q^{R}$ is the same closed WENO5 expression evaluated on the mirrored 5-cell stencil (Shu 1998 §2.2 eq. 2.16) — the algebra is identical, only the input order differs.

Reconstruction (Jiang–Shu 1996)

For face $x_{i+1/2}$ with $U_{i+1/2} > 0$, the left-biased reconstruction is the convex combination

$$q^{L}_{i+1/2} \;=\; \omega_0 p_0 + \omega_1 p_1 + \omega_2 p_2,$$

of three candidate sub-stencil polynomials evaluated at the face (Shu 1998 eq. 2.11):

sub-stencil	offsets	polynomial $p_k(q)$
`p₀`	(−2, −1, 0)	$\tfrac{1}{3}q_{i-2} - \tfrac{7}{6}q_{i-1} + \tfrac{11}{6}q_i$
`p₁`	(−1, 0, +1)	$-\tfrac{1}{6}q_{i-1} + \tfrac{5}{6}q_i + \tfrac{1}{3}q_{i+1}$
`p₂`	(0, +1, +2)	$\tfrac{1}{3}q_i + \tfrac{5}{6}q_{i+1} - \tfrac{1}{6}q_{i+2}$

The nonlinear weights $\omega_k$ are normalized smoothness-weighted versions of the optimal linear weights $(d_0, d_1, d_2) = (1/10, 6/10, 3/10)$:

$$\beta_0 = \tfrac{13}{12}(q_{i-2}-2q_{i-1}+q_i)^2 + \tfrac14(q_{i-2}-4q_{i-1}+3q_i)^2,$$ $$\beta_1 = \tfrac{13}{12}(q_{i-1}-2q_i+q_{i+1})^2 + \tfrac14(q_{i-1}-q_{i+1})^2,$$ $$\beta_2 = \tfrac{13}{12}(q_i-2q_{i+1}+q_{i+2})^2 + \tfrac14(3q_i-4q_{i+1}+q_{i+2})^2,$$ $$\alpha_k = \frac{d_k}{(\varepsilon + \beta_k)^2}, \qquad \omega_k = \frac{\alpha_k}{\sum_j \alpha_j}.$$

Every term above appears in the rule’s replacement AST as +, -, *, /, and ^ over index selectors — there is no smoothness_indicators blob, no nonlinear_weights blob, and no ratio-form ω that has to live outside the AST.

Boundary conditions

Boundary handling is read from the domain’s boundary_conditions block (esm-spec §11.5) at lowering time and applied as downstream rewrite rules over concrete indices — there is no bc:* op embedded in the lowered AST. The pattern is:

Domain BC	Index transformation applied to `$q[$x ± k]`, `$U[$x ± k]`
`periodic`	`mod($x ± k + N, N)` (see `periodic_bc`)
`dirichlet` / `constant`	Boundary cells read the prescribed value via `index` into a fill row
`zero_gradient` / `neumann`	Mirror the in-range neighbor (`min`/`max` clamp on the index)

Each transformation is a separate rule that fires at the boundary cells. The weno5_advection rule itself stays BC-agnostic — its replacement is the interior closed form. The lowering pipeline (rule application + BC rewrites) takes the (grid_family, BC list) pair from the domain and emits index expressions that respect the declared BCs.

Stencil

For a single divergence at cell i, the closed AST reads from cells i−3, …, i+3 (seven cells) for q — five per face — and from cells i−1, i, i+1 for the cell-to-face velocity average. The right-biased branch (used when U_face < 0) is the mirror of the left-biased branch under index reflection.

Convergence

Numeric coverage lives in the canonical Julia test fixture at tests/fixtures/weno5_advection/, exercised by test/test_weno5_advection_rule.jl. The MMS fixture uses f(x) = sin(2πx + 1) on [0, 1] with periodic boundary conditions; the phase shift keeps the critical points of f away from every dyadic cell face at n ∈ {32, 64, 128, 256}, sidestepping the well-known WENO5-JS accuracy dip from ω_k → d_k recovery stalling at f'(x_{i+1/2}) = 0 (Henrick, Aslam & Powers, JCP 2005).

`n`	`dx`	L∞ error	observed order
32	0.03125000	3.4137e-05	—
64	0.01562500	1.0656e-06	5.002
128	0.00781250	3.3254e-08	5.002
256	0.00390625	1.0377e-09	5.002

Theoretical asymptotic order: 5.0 (Jiang & Shu 1996, smooth regions). Acceptance threshold: min(observed order) ≥ 4.7 — leaves headroom for the small accuracy hit from the ε = 1e-6 regularisation of the nonlinear weights. A companion shock-capturing fixture advects a unit square wave one full period at CFL 0.4 with SSP-RK3; max overshoot/undershoot is ~3.7e-4, well under the 0.05 tolerance.

The walker-side fixture under discretizations/finite_volume/weno5_advection/fixtures/convergence/ exercises Layer-B convergence through ESS’s AST-walker dispatch over the closed arrayop + ifelse replacement — no scheme-specific kernels.