On the order of accuracy of finite-volume schemes on unstructured meshes

TL;DR

The paper addresses why high-order finite-volume schemes on unstructured meshes sometimes converge faster than their truncation-order would suggest. By introducing a zero-mean truncation-error condition on $(p+1)$-th order polynomials over mesh periods, it provides a predictive, practical criterion for supra-convergence across several FV schemes, including polynomial-reconstruction, multislope, 1-exact edge-based, and flux-correction methods. The authors prove the zero-mean property for these schemes (under appropriate assumptions) and, with additional conditions, establish sufficiency for $(p+1)$-th order convergence; they also demonstrate this behavior numerically on linear vortices and high-Reynolds-number flows around complex bodies. The work offers a rigorous framework that connects mesh structure, truncation errors, and convergence rates, informing the design of high-order FV methods on unstructured meshes and guiding practitioners in anticipating supra-convergence in simulations.

Abstract

We consider finite-volume schemes for linear hyperbolic systems with constant coefficients on unstructured meshes. Under the stability assumption, they exhibit the convergence rate between $p$ and $p+1$ where $p$ is the order of the truncation error. Our goal is to explain this effect. The central point of our study is that the truncation error on $(p+1)$-th order polynomials has zero average over the mesh period. This condition is verified for schemes with a polynomial reconstruction, multislope finite-volume methods, 1-exact edge-based schemes, and the flux correction method. We prove that this condition is necessary and, under additional assumptions, sufficient for the $(p+1)$-th order convergence. Furthermore, we apply the multislope method to a high-Reynolds number flow and explain its accuracy.

Figures (7)

• Figure 1: Mesh refinement by scaling. Left to right: $h=1$, $h=1/2$, $h=1/4$
• Figure 2: Construction of the numerical flux in the multislope scheme
• Figure 3: Construction of the numerical flux in the multislope scheme on a uniform Cartesian mesh
• Figure 4: Median cell on a 2D triangular mesh
• Figure 5: The use of an auxiliary mesh to prove the zero-mean-error property. Left: mesh in the unit cube; middle: mesh in the cube with edge $m$; right: auxiliary mesh

Theorems & Definitions (48)

• Lemma 6.1
• proof
• Lemma 6.2
• proof
• Proposition 7.1
• proof
• Proposition 7.2
• proof
• Proposition 7.3
• proof