Second-order TVD of the van Leer flux limiter

Demonstrate that the van Leer flux limiter φ_vl(r) = (r + |r|)/(1 + |r|) satisfies the Sweby total variation diminishing criteria, thereby guaranteeing that second-order schemes extrapolated from Lax–Friedrichs or Roe solvers using φ_vl(r) are second-order TVD.

Background

The paper formalizes conditions under which flux limiters yield second-order TVD schemes via Sweby’s criteria. The van Leer limiter is a standard choice widely used in high-resolution schemes.

Although the framework proves symmetry and TVD for other limiters, the authors explicitly state that they fail to find a valid proof that the van Leer limiter satisfies the second-order TVD property. They note this is due to current algebraic limitations in their simplifier, implying a concrete open verification challenge.