A Study of Improved Limiter Formulations for Second-Order Finite Volume Schemes Applied to Unstructured Grids

A general, compact way of achieving second-order in finite-volume numerical methods is to perform a MUSCL-like, piecewise linear reconstruction of flow properties at each cell interface. To avoid the surge of spurious oscillations in the discrete solution, a limiter function is commonly employed. This strategy, however, can add a series of drawbacks to the overall numerical scheme. The present paper investigates this behavior by considering three different limiter formulations in the context of a second-order, finite volume scheme for the simulation of steady, turbulent flows on unstructured meshes. Three limiter formulations are considered: the original Venkatakrishnan limiter, Wang’s modification to the Venkatakrishnan limiter and Nishikawa’s recently introduced R3 limiter. Three different configurations of the fully-developed, two-dimensional, transonic NACA 0012 airfoil are analyzed, configured with different angles of attack and similar freestream properties. The gas dynamics are modeled using the Reynolds-averaged Navier-Stokes (RANS) equations, where the negative Spalart-Allmaras turbulence model is used to solve the closure problem. All limiters are shown to yield similar results for all configurations of this case, although with different dissipative characteristics, provided their control constants are used within appropriate intervals. The presented numerical results are in good agreement with experimental data available in the literature.

💡 Research Summary

The paper investigates the performance of three limiter formulations— the original Venkatakrishnan limiter, Wang’s modification of the Venkatakrishnan limiter, and Nishikawa’s recently introduced R³ limiter— within a second‑order finite‑volume framework on unstructured meshes. The authors implement a MUSCL‑type piecewise‑linear reconstruction for the conserved variables and employ Roe’s flux difference splitting for the inviscid fluxes. Turbulence closure is achieved using the negative Spalart‑Allmaras (SA‑neg) model, and the governing Reynolds‑averaged Navier‑Stokes (RANS) equations are solved implicitly with an Euler time discretization. Linear systems arising from each pseudo‑time step are solved with restarted GMRES (restart after 200 iterations) preconditioned by an Additive Schwarz method combined with an ILU(3) factorization on each subdomain.

Three test cases are considered: a transonic NACA 0012 airfoil at three angles of attack (0°, +0.96°, +2.03°) with freestream Reynolds numbers around 6 × 10⁶ and Mach numbers near 0.8. The same C‑shaped unstructured hexahedral mesh (17 920 × 512 cells, y⁺ < 1 near the wall) is used for all cases, and standard boundary conditions (adiabatic wall on the airfoil, non‑reflecting far‑field) are applied. The study focuses on the ability of each limiter to capture shock waves without spurious oscillations, to maintain low artificial dissipation, and to achieve robust convergence.

The Venkatakrishnan limiter computes a cell‑wise limiter value based on the minimum and maximum of the variable among the cell and its immediate neighbors, with an ε² term proportional to the cube of a characteristic cell length (εV Δx)³. This term prevents division‑by‑zero problems in nearly constant regions but can become overly restrictive on highly non‑uniform meshes. Wang’s modification replaces the local ε² with a global formulation based on the overall maximum and minimum of the variable, scaled by a constant εW. This change reduces sensitivity to mesh size variation and improves robustness on irregular grids.

Nishikawa’s R³ limiter belongs to the Rᵖ family (p = 3) and uses a cubic combination of the forward and backward differences (Δ⁺, Δ⁻) together with an ε³ term (εR³ Δx)⁴. Although theoretically capable of preserving third‑order accuracy in smooth regions, the underlying numerical scheme in this work is only second‑order, so the R³ limiter mainly contributes additional dissipation control. In practice, the R³ limiter exhibits the lowest artificial viscosity among the three, leading to sharper shock resolution and more accurate pressure and temperature gradients.

Convergence behavior is examined by ramping the global CFL number from 0.1 to 10 000 while monitoring the residual of the continuity equation. For the first two configurations (0° and +0.96°), all three limiters achieve the prescribed convergence criterion (a reduction of six orders of magnitude in the continuity residual). However, for the third configuration (+2.03°), all limiters experience stagnation, failing to reduce the residual below the target. The authors attribute this to the combination of a high CFL number and the limited piecewise‑linear reconstruction, which together hinder the implicit Jacobian from capturing the full non‑linear behavior of the limited scheme. When the limiter is set to zero (effectively a first‑order scheme), convergence to machine zero is possible but extremely slow. Conversely, removing the limiter entirely (ψ = 1) leads to severe non‑physical oscillations around the shock and rapid divergence.

Parameter sensitivity studies reveal that acceptable ranges for the control constants are εV ∈