[Paper Review] Hydrodynamic simulations on a moving Voronoi mesh

Volker Springel|arXiv (Cornell University)|Sep 10, 2011

Fluid Dynamics Simulations and InteractionsEngineering2 references21 citations

TL;DR

This paper introduces a novel moving Voronoi mesh method for hydrodynamic simulations that uses Voronoi tessellation of movable mesh-generating points to achieve high-accuracy, Galilean-invariant, and distortion-free Lagrangian hydrodynamics. The approach combines second-order finite volume Godunov schemes with exact Riemann solvers on unstructured meshes, demonstrating superior accuracy and convergence at lower resolution than fixed Eulerian or SPH methods, especially in problems with shocks, contact discontinuities, and bulk flows.

ABSTRACT

At the heart of any method for computational fluid dynamics lies the question of how the simulated fluid should be discretized. Traditionally, a fixed Eulerian mesh is often employed for this purpose, which in modern schemes may also be adaptively refined during a calculation. Particle-based methods on the other hand discretize the mass instead of the volume, yielding an approximately Lagrangian approach. It is also possible to achieve Lagrangian behavior in mesh-based methods if the mesh is allowed to move with the flow. However, such approaches have often been fraught with substantial problems related to the development of irregularity in the mesh topology. Here we describe a novel scheme that eliminates these weaknesses. It is based on a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points. The mesh is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite volume approach, based on a second-order Godunov scheme with an exact Riemann solver. A particularly powerful feature of the approach is that the mesh-generating points can in principle be moved arbitrarily. If they are given the velocity of the local flow, a highly accurate Lagrangian formulation of continuum hydrodynamics is obtained that is free of mesh distortion problems, while it is at the same time fully Galilean-invariant, unlike ordinary Eulerian codes. We describe the formulation and implementation of our new Voronoi-based hydrodynamics, and we discuss a number of illustrative test problems that highlight its performance in practical applications.

Motivation & Objective

• To address fundamental limitations in existing hydrodynamical methods, such as poor shock resolution in SPH and lack of Galilean invariance in Eulerian codes.
• To develop a mesh-based method that achieves Lagrangian behavior without mesh tangling or topological artifacts.
• To combine the geometric flexibility of SPH with the accuracy and conservation properties of mesh-based methods.
• To enable adaptive resolution and accurate simulation of astrophysical flows with strong density contrasts and bulk motions.

Proposed method

• The mesh is constructed as the Voronoi tessellation of a set of movable points, which continuously deform with the flow to maintain mesh quality.
• Hydrodynamics is solved using a second-order accurate finite volume Godunov scheme with an exact Riemann solver for hyperbolic conservation laws.
• The mesh-generating points are moved with the local fluid velocity to achieve a nearly Lagrangian formulation, minimizing advection errors.
• The method ensures topological robustness because Voronoi tessellations inherently avoid mesh tangling or bow-tie cells.
• The approach is fully Galilean-invariant due to the mesh motion following the fluid, eliminating velocity-dependent numerical artifacts.
• The scheme supports adaptive resolution and curved boundary conditions naturally through the unstructured mesh framework.

Experimental results

Research questions

• RQ1 Can a moving mesh method based on Voronoi tessellation achieve Galilean invariance and avoid mesh tangling in multi-dimensional hydrodynamics?
• RQ2 How does the accuracy of the moving Voronoi mesh compare to fixed Eulerian and SPH codes in standard hydrodynamical test problems?
• RQ3 To what extent does the moving mesh reduce advection errors and improve convergence in simulations with bulk flows or contact discontinuities?
• RQ4 Can the method maintain high accuracy in problems with strong density contrasts and complex flow morphologies, such as the Kelvin-Helmholtz instability or cloud crushing?

Key findings

• The moving Voronoi mesh achieves converged results in the cloud crushing test at lower resolution than fixed-mesh Eulerian codes, indicating superior numerical efficiency.
• The moving-mesh method consistently predicts a higher residual cloud mass fraction than the fixed-mesh code, due to reduced advection errors and Galilean invariance.
• The method shows improved shock and contact discontinuity resolution compared to SPH, which underestimates mass loss in cloud disruption simulations.
• The code demonstrates second-order accuracy in space and time, with no need for artificial viscosity, and avoids spurious surface tension effects common in SPH.
• In 3D, the code is about 2× slower than SPH and 3–4× slower than fixed-mesh Eulerian codes per resolution element, but this cost is offset by higher accuracy at lower resolution.
• The method is robust in complex flows, including those with curved boundaries and strong bulk motions, due to its geometric flexibility and topological stability.