Semi-implicit Lagrangian Voronoi Approximation for the incompressible Navier-Stokes equations

TL;DR

The paper presents SILVA, a Semi-Implicit Lagrangian Voronoi Approximation for incompressible flows that uses explicit mesh motion and a global implicit pressure projection on a moving Voronoi tessellation to enforce $\nabla\cdot\mathbf{v}=0$. It introduces a robust discrete gradient operator on irregular Voronoi meshes, a semi-discrete and semi-implicit time integration framework, and a stabilization strategy to mitigate vortex-core instabilities, yielding a sparse, symmetric Poisson-like system for pressure. Validation on Taylor-Green, Gresho, lid-driven cavity, and Rayleigh–Taylor benchmarks demonstrates accuracy, stability, and efficient boundary handling, highlighting advantages over ISPH in matrix sparsity and boundary condition implementation. The method supports topology-changing meshes without remapping and shows promise for multi-phase and fluid-structure interaction applications, with clear paths for extension to higher order accuracy, 3D problems, and compressible flows.

Abstract

We introduce Semi-Implicit Lagrangian Voronoi Approximation (SILVA), a novel numerical method for the solution of the incompressible Euler and Navier-Stokes equations, which combines the efficiency of semi-implicit time marching schemes with the robustness of time-dependent Voronoi tessellations. In SILVA, the numerical solution is stored at particles, which move with the fluid velocity and also play the role of the generators of the computational mesh. The Voronoi mesh is rapidly regenerated at each time step, allowing large deformations with topology changes. As opposed to the reconnection-based Arbitrary-Lagrangian-Eulerian schemes, we need no remapping stage. A semi-implicit scheme is devised in the context of moving Voronoi meshes to project the velocity field onto a divergence-free manifold. We validate SILVA by illustrative benchmarks, including viscous, inviscid, and multi-phase flows. Compared to its closest competitor, the Incompressible Smoothed Particle Hydrodynamics (ISPH) method, SILVA offers a sparser stiffness matrix and facilitates the implementation of no-slip and free-slip boundary conditions.

Figures (13)

• Figure 1: A Voronoi cell for the green particle in the center can be obtained by intersecting $N-1$ half-planes, one for each other seed. The cell list structure optimizes the computation by focusing on nearby seed only. The orange and blue colors highlight the partition $P_{i,k}$ and the incomplete Voronoi cell $\omega_i^k$ respectively for $k = 0, 1, \dots, 5$.
• Figure 2: The magnitude of $\nabla f$ on a square $\Omega=[0, 1]^2$ with an irregular Voronoi grid of 6400 cells (a) for $f = \frac{1}{\pi} \cos (\pi x) \cos (\pi y)$. The image (b) and (c) show exact solution and the strong gradient \ref{['eq:sgrad']}, respectively, and they are barely distinguishable. The image (d) uses the weak gradient \ref{['eq:wgrad']} and is infested with spurious noise.
• Figure 3: Example of a sparsity pattern of the left-hand-side matrix in SILVA and ISPH on uniformly random distribution of 625 points (Voronoi seeds) in two dimensions. The smoothing length for ISPH is $1.5\delta r$ and the Wendland quintic kernel is used. For the sake of readability, the points are sorted by their occurrence in the cell list.
• Figure 4: Errors of velocity, pressure, energy and velocity divergence in the Taylor-Green vortex benchmark. The horizontal axis corresponds to $\log_{10} N$, where $N$ is the number of cells per side of the computational domain. The vertical axis indicates the logarithm of error. We also compare the result for a structured (ST) rectangular and an unstructured (UN) Vogel grid for $\mathrm{Re}=\{ 400, 1000, \infty\}$. The dashed triangles indicate what the reference linear and quadratic convergence slopes.
• Figure 5: The final time step of Taylor-Green Vortex for $N = 16$, $\mathrm{Re} = \infty$.

Theorems & Definitions (14)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 3
• proof