flux_limiter_superbee

Family
limiter
Grid family
cartesian
Kind
limiter
Accuracy
O(dx²) in smooth monotone regions; compressive near discontinuities
Applies to
limit(r)
Rule file
discretizations/finite_volume/flux_limiter_superbee.json
Tags
#limiter #tvd #superbee #compressive

Limiter curve

Superbee limiter φ(r) and Sweby second-order TVD region
Superbee φ(r) overlaid on the Sweby (1984) second-order TVD region. The curve hugs the upper edge — at r = 1 it is twice the identity (slope 2 from the origin), then saturates at 2 — which is what makes the scheme compressive: it amplifies sharp gradients but degrades smooth extrema more aggressively than minmod.

The rule is a scalar AST in the slope-ratio variable $r; it carries no spatial stencil of its own. The caller computes r at each interface and multiplies the high-order slope correction by φ(r) — see the discrete form below and the worked example in discretizations/finite_volume/README.md.

Discrete form

Slope ratio at cell interface i + ½ (1D, uniform Cartesian, u > 0):

$$r_i \;=\; \frac{q_i - q_{i-1}}{q_{i+1} - q_i + \varepsilon}$$

(small ε guards the locally-flat case). The superbee limiter is

$$\varphi(r) \;=\; \max\!\bigl(0,\;\min(2r,\,1),\;\min(r,\,2)\bigr).$$

A MUSCL-style high-order flux scales the slope correction by φ:

$$F_{i+\frac{1}{2}} \;=\; u\,\Bigl(q_i \;+\; \tfrac{1}{2}\,\varphi(r_i)\,(q_{i+1} - q_i)\Bigr).$$

Under forward Euler with CFL ≤ 1 / (1 + 0.5·φ_max) (φ_max = 2 for superbee, so CFL ≤ 0.5), this scheme is strictly TVD. The composition is reconstruction-agnostic: the same φ(r) factor applies when pairing with PPM or WENO-5 reconstructions.

PropertyValue
TVDyes
Monotonicity-preservingyes — φ(r) = 0 for r ≤ 0
Sweby upper boundφ(r) ≤ 2
Consistencyφ(1) = 1
Symmetricyes — φ(r)/r = φ(1/r)
Smooth-extremum behaviorcompressive (steepens)
φ_max2

References: Roe (1986), Ann. Rev. Fluid Mech. 18:337–365, eq. (36); Sweby (1984), SIAM J. Numer. Anal. 21(5):995–1011, fig. 4.

Convergence

Convergence plot pending fixture activation. The convergence fixture at [discretizations/finite_volume/flux_limiter_superbee/fixtures/convergence/](https://github.com/EarthSciML/EarthSciDiscretizations/blob/main/discretizations/finite_volume/flux_limiter_superbee/fixtures/convergence) currently declares applicable: false: a slope-ratio limiter is a scalar AST, not a stencil, and its acceptance criterion is monotonicity preservation (Sweby region, φ(1) = 1, strict TVD on slope-ratio inputs), not asymptotic convergence-order on a manufactured solution. The Layer-B′ monotonicity / TVD harness lands the matching fixture kind; until then the [fixtures/monotonicity/](https://github.com/EarthSciML/EarthSciDiscretizations/blob/main/discretizations/finite_volume/flux_limiter_superbee/fixtures/monotonicity) sweep — Sweby-region properties verified directly off the rule's AST plus a CFL-bounded TVD check on a smooth + square-wave initial condition — is the authoritative numeric coverage for this rule.

See also