A node-conservative vorticity-preserving Finite Volume method for linear acoustics on unstructured grids

TL;DR

This work introduces a truly multi-dimensional, node-centered conservation strategy for finite-volume discretizations of linear acoustics on unstructured grids. By employing non-standard Riemann solvers with free parameters and enforcing nodal conservation, the authors derive a vorticity-preserving method on general meshes, including a novel nodal-pressure formulation that yields a discrete gradient naturally coupled to a node-based divergence. On Cartesian grids, the scheme reduces to known discrete gradient and divergence operators, while on unstructured grids it achieves structure preservation through the node-wise flux closure. Second-order extension via least-squares reconstruction maintains vorticity preservation for grids with suitable cell topology, and numerical experiments confirm accurate wave propagation, robust stationary-vortex behavior, and improved coarse-grid fidelity. The results offer a principled path toward structure-preserving schemes beyond Cartesian geometries and set the stage for 3D extensions and broader hyperbolic systems.

Abstract

Instead of ensuring that fluxes across edges add up to zero, we split the edge in two halves and also associate different fluxes to each of its sides. This is possible due to non-standard Riemann solvers with free parameters. We then enforce conservation by making sure that the fluxes around a node sum up to zero, which fixes the value of the free parameter. We demonstrate that for linear acoustics one of the non-standard Riemann solvers leads to a vorticity preserving method on unstructured meshes.

Figures (18)

• Figure 1: Illustration of conservation: The change in mass in $\Omega_1$ due to the flux through $\gamma$ (dashed) is the negative of the change of mass in $\Omega_2$.
• Figure 2: The dual cell $c_n$ associated to a node $n$ (bounded by the dashed line that joins cell centroids and edge midpoints).
• Figure 3: Different conservation concepts. Left: Local conservation focusing on an edge. Right: Local conservation focusing on a node.
• Figure 4: Left: The definition of L/R depends on the choice of the normal vector $\mathbf n_e$ on edge $e$. If outward normals with respect to cell $c$ are used, then cell $c$ is always L. Right: The node normal.
• Figure 5: Top: Illustration and motivation for the proof of Lemma \ref{['lemma:area']}. Bottom: Notation for the proof of Theorem \ref{['thm:curl']} (the case of $n \neq m$, but $n$ and $m$ sharing an edge $e$).

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Definition 4.1
• Definition 4.2
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof