Symmetry of the superbee flux limiter

Prove that the superbee flux limiter φ_sb(r) = max(0, min(2r, 1), min(r, 2)) is symmetric, i.e., establish φ_sb(r)/r = φ_sb(1/r) for all r > 0, thereby ensuring that the limiter acts equivalently on forward and backward gradients in second-order finite volume reconstructions.

Background

Second-order spatial accuracy is obtained via flux extrapolation controlled by limiters, with symmetry and TVD properties characterized in Theorems presented earlier in the paper. The authors implement and verify several standard limiters within their pipeline.

They report successful formal proofs for the minmod and monotonized-centered limiters but explicitly state that their theorem-prover fails to find valid proofs for the symmetry property of the superbee limiter. Although the authors believe this property holds unconditionally, it remains unproven in their formal system, motivating a focused verification effort.