weno5_advection

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

Stencil

Coefficients (left-biased)

sub-stencil	offsets	coefficients
`p₀`	(−2, −1, 0)	(1/3, −7/6, 11/6)
`p₁`	(−1, 0, +1)	(−1/6, 5/6, 1/3)
`p₂`	(0, +1, +2)	(1/3, 5/6, −1/6)

Optimal linear weights (d₀, d₁, d₂) = (1/10, 6/10, 3/10). Smoothness indicators (Jiang & Shu 1996 eq. 2.17):

$$ \beta_0 = \tfrac{13}{12}(q_{i-2}-2q_{i-1}+q_i)^2 + \tfrac14(q_{i-2}-4q_{i-1}+3q_i)^2, $$ $$ \beta_1 = \tfrac{13}{12}(q_{i-1}-2q_i+q_{i+1})^2 + \tfrac14(q_{i-1}-q_{i+1})^2, $$ $$ \beta_2 = \tfrac{13}{12}(q_i-2q_{i+1}+q_{i+2})^2 + \tfrac14(3q_i-4q_{i+1}+q_{i+2})^2. $$

Nonlinear weights α_k = d_k / (ε + β_k)², normalized to ω_k = α_k / Σ α_j. The right-biased branch (used when u < 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.

Pending ESS walker harness extension. The Layer-B walker harness needs nonlinear-reconstruction support — the ratio-form nonlinear weights `ω_k = d_k / (ε + β_k)²` are not yet expressible in the §7 stencil schema. The walker-side fixture under [discretizations/finite_volume/weno5_advection/fixtures/convergence/](https://github.com/EarthSciML/EarthSciDiscretizations/blob/main/discretizations/finite_volume/weno5_advection/fixtures/convergence) records a structured SKIP and will populate once that extension lands; until then the canonical numeric coverage above is the source of truth.