Numerical investigations of cosmological spacetimes can be categorized into two broad classes of calculations, distinguished by their computational goals: (i) geometrical and mathematical principles of cosmological models, and (ii) physical and astrophysical cosmology. In the former, the emphasis is on the geometric framework in which astrophysical processes occur, for example cosmological expansion, topological singularities, geometrodynamics in general, and classification characteristics or invariants of the many models allowed by the theory of general relativity. In the latter, the emphasis is on the cosmological and astrophysical processes in the real or observable Universe, and the quest to determine the model which best describes our Universe. The former is pure in the sense that it concerns the fundamental nonlinear behavior of the Einstein equations and the gravitational field. The latter is more complex as it addresses the composition, organization, and dynamics of the Universe from the small scales (fundamental particles and elements) to the large (galaxies and clusters of galaxies). However the distinction is not always so clear, and geometric effects in the spacetime curvature can have significant consequences for the evolution and observation of matter distributions.

Any comprehensive model of cosmology must therefore include nonlinear interactions between different matter sources and spacetime curvature. A realistic model of the Universe must also cover large dynamical spatial and temporal scales, extreme temperature and density distributions, and highly dynamic atomic and molecular matter compositions. In addition, due to all the varied physical processes of cosmological significance, one must draw from many disciplines of physics to model curvature anisotropies, gravitational waves, electromagnetic fields, nucleosynthesis, particle physics, hydrodynamic fluids, etc. These phenomena are described in terms of coupled nonlinear partial differential equations and must be solved numerically for general inhomogeneous spacetimes. The situation appears extremely complex, even with current technological and computational advances. As a result, the codes and numerical methods that have been developed to date are designed to investigate very specific problems with either idealized symmetries or simplifying assumptions regarding the metric behavior, the matter distribution/composition or the interactions among the matter types and spacetime curvature.

It is the purpose of this article to review published numerical cosmological calculations addressing problems from the very early Universe to the present; from the purely geometrical dynamics of the initial singularity to the large scale structure of the Universe. There are three major sections: Section 2 where a brief overview is presented of various defining events occurring throughout the history of our Universe and in the context of the standard model, Section 3 where reviews of early Universe and relativistic cosmological calculations are presented, and Section 4 which focuses on structure formation in the post-recombination epoch and on testing cosmological models against observations. Following the summary paragraphs in Section 5, an appendix in Section 6 presents the basic Einstein equations, kinematic considerations, matter source equations with curvature, and the equations of perturbative physical cosmology on background isotropic models. References to numerical methods are also supplied and reviewed for each case.

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