Semi‐implicit Lagrangian Voronoi approximation for the incompressible Navier–Stokes equations

• Published in: International journal for numerical methods in fluids 2025-01, Vol.97 (1), p.88-115
• Main Authors: Kincl, Ondřej, Peshkov, Ilya, Boscheri, Walter
• Format: Article
• Language: English
• Subjects: Approximation; Benchmarks; Boundary conditions; Computational grids; Divergence; Fluid flow; Hydrodynamics; Incompressible flow; incompressible flows; Lagrangian Voronoi meshes; Mathematical models; Mathematics; mesh regeneration with topology changes; Navier-Stokes equations; Numerical methods; particle hydrodynamics; semi‐implicit schemes; Smooth particle hydrodynamics; Stiffness matrix; Time dependence; Time marching; Topology; Velocity; Velocity distribution; Voronoi graphs
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5339

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 method, SILVA offers a sparser stiffness matrix and facilitates the implementation of no‐slip and free‐slip boundary conditions. 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 semi‐implicit time marching with time‐dependent Voronoi tessellations with topology changes. In SILVA, the numerical solution is stored at particles, which move with the fluid velocity and play the role of the generators of the computational mesh. The velocity field is projected onto a divergence‐free manifold. We validate SILVA by illustrative benchmarks, including viscous, inviscid, and multiphase flows.