<p>Self-organization, characterized by the ordered temporal and spatial evolution of events and structures, is a fundamental phenomenon in the cosmos. Theoretical studies of structure formation in the Universe heavily rely on modeling, as it often represents the only viable approach to validate theoretical predictions, particularly in the context of gravitational gas dynamics with its inherent nonlinearity. This holds true for the intricate processes involved in the development of hierarchical structures, entailing complex alterations in spatial geometry. Consequently, modeling has yielded a vast array of profound astrophysical insights and discoveries. A pivotal aspect of modeling gas dynamics and hydrodynamics is the mathematical representation of the fundamental conservation laws governing continuous media, encompassing mass, momentum, and total energy</p>