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The theory and application of numerical methods for unstructured meshes havebeen improved significantly in recent years. Because the grids can be placearbitrarily in space, unstructured meshes can provide much higher spatialresolution than regular meshes. The built-in nature of mesh adaptivity forunstructured meshes gives one way to simulate highly dynamic, hierarchicalproblems involving both collisionless dark matter and collisional gas dynamics.In this paper, we describe algorithms to construct unstructured meshes from aset of points with periodic boundary conditions through Delaunay triangulation,and algorithms to solve hydrodynamic and N-body problems on an unstructuredmesh . A combination of a local transformation algorithm and the traditionalBowyer-Watson algorithm gives an efficient approach to perform Delaunaytriangulation. A novel algorithm to solve N-body equations of motion on anunstructured mesh is described. Poisson's equation is solved using theconjugate gradient method. A gas-kinetic scheme based on the BGK model to solveEuler equations is used to evolve the hydrodynamic equations. We apply thesealgorithms to solve cosmological settings, which involve both dark and baryonicmatter. Various cooling and heating processes for primordial baryonic matterare included in the code. The numerical results show that the N-body andhydrodynamic algorithms based on unstructured meshes with mesh refinement arewell-suited for hierarchical structure formation problems.

Hydrodynamic and N-body schemes on an unstructured, adaptive mesh with applications to cosmological simulations