E pur si muove: Galiliean-invariant cosmological hydrodynamical simulations on a moving mesh (0901.4107v3)

Abstract: Hydrodynamic cosmological simulations at present usually employ either the Lagrangian SPH technique, or Eulerian hydrodynamics on a Cartesian mesh with adaptive mesh refinement. Both of these methods have disadvantages that negatively impact their accuracy in certain situations. We here propose a novel scheme which largely 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 unsplit Godunov scheme with an exact Riemann solver. The mesh-generating points can in principle be moved arbitrarily. If they are chosen to be stationary, the scheme is equivalent to an ordinary Eulerian method with second order accuracy. If they instead move with the velocity of the local flow, one obtains a Lagrangian formulation of hydrodynamics that does not suffer from the mesh distortion limitations inherent in other mesh-based Lagrangian schemes. In this mode, our new method is fully Galilean-invariant, unlike ordinary Eulerian codes, a property that is of significant importance for cosmological simulations. In addition, the new scheme can adjust its spatial resolution automatically and continuously, and hence inherits the principal advantage of SPH for simulations of cosmological structure growth. The high accuracy of Eulerian methods in the treatment of shocks is retained, while the treatment of contact discontinuities improves. We discuss how this approach is implemented in our new parallel code AREPO, both in 2D and 3D. We use a suite of test problems to examine the performance of the new code and argue that it provides an attractive and competitive alternative to current SPH and Eulerian techniques. (abridged)

• The paper introduces a novel moving-mesh simulation method that preserves Galilean invariance, significantly improving cosmological hydrodynamic modeling.
• The method employs a moving Voronoi tessellation combined with a second-order Godunov scheme, effectively resolving shocks and discontinuities.
• The implementation in AREPO demonstrates enhanced handling of fluid instabilities and supersonic flows, paving the way for high-fidelity cosmic structure simulations.

Galilean-Invariant Cosmological Hydrodynamical Simulations on a Moving Mesh

The paper by Volker Springel introduces a novel hydrodynamic simulation technique designed to ameliorate some of the inherent limitations in traditional cosmological hydrodynamics methods, specifically those related to the Lagrangian smoothed particle hydrodynamics (SPH) and Eulerian mesh-based adaptive mesh refinement (AMR) codes. These traditional methods face challenges with certain fluid dynamics phenomena, such as fluid instabilities and Galilean invariance, which can greatly impact the accuracy of simulations in supersonic flow regimes typical in cosmological contexts. This paper proposes a sophisticated numerical scheme using a moving Voronoi tessellation mesh, offering an efficient and more accurate alternative for simulating the hydrodynamics of cosmological structure formation.

Methodological Framework

The core of the method utilizes a moving unstructured mesh defined by a Voronoi tessellation of discrete points, integrating a second-order Godunov scheme with an exact Riemann solver. The distinct feature of this approach is the flexibility in the movement of mesh-generating points, allowing arbitrary movement - stationary for equivalence to traditional Eulerian methods or movement with local fluid velocity which provides Galilean invariance. This capacity is particularly essential in cosmological simulations where supersonic flows are ubiquitous.

Key Innovations and Improvements

The moving mesh approach achieves a significant advancement with its Galilean invariance property. Unlike SPH, which suffers from suppressed fluid instabilities due to poor resolution and incorrect treatment of discontinuities, and AMR, which faces issues with reference frames and high-speed flows, the proposed method provides a fully Galilean-invariant framework. This effectively mitigates problematic reference frame dependencies inherent in Eulerian codes. Additionally, it preserves the high accuracy of Eulerian methods in handling shocks while improving the treatment of contact discontinuities, addressing a notable gap with SPH methods.

The paper details the implementation of this approach within a new code called AREPO. AREPO supports 2D and 3D simulations, offering parallelization for distributed memory architectures, a key requirement for modern large-scale simulations. The code integrates modular features for dynamic mesh refinement, allowing adaptive spatial resolution adjustments without the abrupt shifts typically inherent in AMR frameworks.

Numerical Results and Performance Evaluation

The effectiveness of AREPO is demonstrated through astute numerical studies and test problems, including classical benchmarks like shock tube tests, Sedov blast waves, and Kelvin-Helmholtz instabilities. Particularly notable is the adept handling of cold flows in cosmological simulations without introducing significant spurious heating, a common pitfall in both SPH and AMR techniques. Additionally, AREPO can dynamically adjust the mesh, limiting resolution loss and overmixing risks, and maintaining mathematical rigor in the simulation outputs over vast temporal and spatial scales characteristic of cosmological studies.

Implications and Potential

The proposed moving-mesh hydrodynamic code aligns well with the demands of modern cosmology, offering improved stability, reduced numerical diffusion, and enhanced computational efficiency. It addresses critical issues of reference frame dependence and resolves small-scale instabilities better than its AMR and SPH counterparts. This advancement opens new pathways for investigating astrophysical phenomena where high precision and dynamic range are crucial. The work encourages reconsiderable layers in simulator design that can leverage adaptive and geometric flexibility to optimize computational astrophysics research.

In conclusion, the moving mesh paradigm Springel develops signifies a substantial methodological leap over traditional approaches in cosmological hydrodynamics. As computational capabilities advance, such versatile frameworks that blend the robustness of classical methods with innovative adaptive strategies represent the frontier for high-fidelity simulations of cosmic structure formation. The implications for both theoretical understandings and observational interpretations in cosmology are profound and merit further exploration in future research.