weno5_advection_2d
Stencil
(i,j). Each axis carries the three candidate sub-stencils of
the 1D rule. The union spans nine cells per axis (five own-axis +
four cross-axis), 17 of the 5×5 neighborhood total.Coefficients
Per-axis blocks under axes.x / axes.y reuse the 1D
weno5_advection
sub-stencils, linear weights (d₀, d₁, d₂) = (1/10, 6/10, 3/10), and
smoothness indicators (Jiang & Shu 1996 eq. 2.17), with the selector axis
swapped between $x and $y. 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 the
local face velocity is negative) is the index-reflected mirror of the
left-biased branch.
The flux selection per axis is upwinded against the local face velocity:
$$ F^x_{i+1/2,j} = u_{i+1/2,j}\, q_{i+1/2,j}^{\mathrm{upwind}},\qquad F^y_{i,j+1/2} = v_{i,j+1/2}\, q_{i,j+1/2}^{\mathrm{upwind}}. $$The advective tendency sums the two axis flux divergences:
$$ \mathrm{advect}(q, U) = \frac{F^x_{i+1/2,j} - F^x_{i-1/2,j}}{\Delta x} \,+\, \frac{F^y_{i,j+1/2} - F^y_{i,j-1/2}}{\Delta y}. $$Convergence
u(x,y) = sin(2π x + 1)·sin(2π y + 1/2) on
[0,1]² periodic, reconstructed face values vs the analytic
cross-section averages. Slope ≈ 5.0; sub-5th order is recovered in the
asymptotic regime.The numeric coverage lives in two fixtures:
Layer-B canonical convergence —
discretizations/finite_volume/weno5_advection_2d/fixtures/convergence/— declaresexpected_min_order = 4.5onn ∈ {32, 64, 128, 256}with the phase-shifted 2D sine MMS, matching the floor cited by Shu (1998) §2.2 with a small allowance for the ε-regularised nonlinear-weight transition.Layer-C integration convergence —
discretizations/finite_volume/weno5_advection_2d/fixtures/integration/— runs the dimension-by-dimension 1D WENO5 reconstruction along both axes on cell-averaged 2D phase-shifted sine overn ∈ {16, 32, 64, 128}and assertsobserved_min_order ≥ 4.4. Empirically the floor sits at≈ 4.98on this sequence.
n | h | L∞ error | observed order |
|---|---|---|---|
| 16 | 0.06250000 | 1.0733e-03 | — |
| 32 | 0.03125000 | 3.4081e-05 | 4.977 |
| 64 | 0.01562500 | 1.0643e-06 | 5.001 |
| 128 | 0.00781250 | 3.3249e-08 | 5.000 |
Theoretical asymptotic order: 5.0 per axis (Jiang & Shu 1996, smooth
regions; Shu 1998 §2.2 dimension-by-dimension splitting; LeVeque 2002 §20).
Acceptance threshold: min(observed order) ≥ 4.4 across the
n ∈ {16, 32, 64, 128} sweep, with NO acceptance fudging — the
case fails (not skips) if the observed order falls below 4.4 (dsc-5od §5).
mms_convergence routes WENO5 rules unconditionally
to a 1D path (mms_weno5_convergence) that expects
reconstruction_left_biased at the spec root. The 2D rule
carries the same blocks under axes.x / axes.y per
the dimension-by-dimension splitting (Shu 1998), so the existing path
cannot read it. Until ESS gains a 2D dispatch (a follow-up
esm-* bead is needed), Layer B SKIPs with
applicable: false and Layer C is the operational verifier of
the asymptotic 2D order.