Centroidal Voronoi Diagrams and Euler-Lagrangian Methods for Astrophysical Simulations

Large scale cosmological simulations such as galaxy and star formation are of great interest to cosmologists. These simulations are done by consideration of relativistic, compressible, discontinuous fluids. Many techniques from particle based to adaptive meshing have been explored, each with advantages and disadvantages. Here, the limitations of centroidal Voronoi tessellations with these astrophysical fluids are investigated in great detail, showing that centroidal Voronoi tessellations are suitable for use with these fluids. Furthermore, the introduction of an ad hoc time dependence technique for these centroidal Voronoi tessellations is analyzed and shown to be not useful for large scale simulation. Lastly, a new meshing technique is introduced called the Euler-Lagrange centroidal Voronoi tessellations, resembling standard centroidal Voronoi tessellations but enforcing a centroidal Voronoi tessellations that advects with the fluid. 1 Background 1.1 Motivation Knowing how matter has evolved through time is a major part of the field of cosmology. Many interesting phenomena occur in the early universe, such as star or galaxy formation. [8] Cosmological parameters of the universe can be discovered by comparison of the model to observation. Therefore, accurate cosmological simulations of matter through long periods of time must be conducted. [1] Due to the complexity of these cosmological systems, often numerical methods techniques are an indispensable tool in correctly modelling their evolution. [7] Many of the dynamic systems of these simulations are astrophysical gas systems involving the motion of compressible fluid with relativistic velocity or energy densities. [6] Therefore, it is a relevant topic to consider how to simulate relativistic, highly compressible, discontinuous fluids in astrophysics. However, the majority of the work in fluid simulation relates to incompressible fluids. An exploration of past techniques in fluid simulations and their application to the fluids in consideration is done here, along with the specification of some new techniques for the simulation of relativistic, highly compressible, discontinuous fluids. 1.2 Lagrangian Simulations Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics (SPH) is one of the most commonly used techniques for simulating astrophysical fluids. This method works by subdividing the fluid density distribution into discrete pieces, called particles. These particles have a spatial distance of influence, created by smoothing the fluid using some interpolation function, and referred to as the kernel. This means that the physical properties of any SPH particle can be determined by using the kernel function on the relevant properties of all the particles which lie within the range of the kernel. [7]