Numerical methods in high energy theoretical physics and astrophysics

This thesis develops and applies numerical methods to problems in high-energy theoretical physics where strong coupling and nonlinear dynamics limit analytic control. Starting in the framework of QCD, motivated by the elusive phenomena of confinement and screening, the phase structure and the possible presence of first-order transitions, the work adopts holography as a controlled ``theoretical laboratory'' to study confinement, screening, phase structure, and nonequilibrium dynamics in QCD-like models. On the methodological side, the thesis builds a practical toolkit for solving extremization problems and evolution equations, with emphasis on discretizations, quadrature, spline-based approximations in one and two dimensions, and constrained optimization strategies tailored to observables in variational problems. In particular, one- and two-dimensional optimization algorithms were written in the Julia and Matlab languages to tackle these problems; we report the main functions in the thesis. More broadly, the thesis emphasizes how the numerical viewpoint clarifies the interplay between physical observables and the geometric structure of the holographic problem: quantities such as Wilson loops and entanglement entropy become well-posed minimization tasks, while dynamical phenomena translate into constrained evolutions. This perspective makes it possible to access regimes where analytic approaches are impractical, and to extract diagnostics of phase transitions directly from the computed solutions. These techniques are deployed in three main physics applications. First, confinement and screening are investigated through holographic Wilson loops in top-down quiver gauge theories. Second, deconfinement transitions are probed via holographic entanglement entropy in the same class of quiver theories, including minimal-surface embeddings that couple spatial and internal directions. Third, a bottom-up Einstein–Maxwell–dilaton model is used to construct a QCD-like phase diagram to study bubble dynamics and real-time evolution across the first-order phase transition, complemented by thermodynamic and stability diagnostics.