The lattice Boltzmann method (LBM) has become a powerful tool in computational fluid dynamics, yet its application to high-speed compressible flows remains a significant challenge. Considering all the developments towards achieving this goal, the numerical equilibrium approach stands out. This thesis presents a comprehensive framework for simulating compressible flows across subsonic, transonic, and supersonic regimes, addressing long-standing limitations in Numerical Equilibrium. Through an introduction of novel techniques, this work bridges gaps between mathematical theory and practical applications, offering robust solutions for high-speed compressible and multiphase flow dynamics. The first contribution redefines the collision operator by framing it as a linear algebra problem, cre- ating a unified framework that encompasses both linear and nonlinear methods. In this context, the linear approach refers to polynomial expansions, while the nonlinear approach pertains to the non-linear moment-matching technique introduced in numerical equilibrium [66]. The second contribution intro- duces a novel collision operator that extends the nonlinear moment-matching methodology aptly named, numerical collision. This innovation replaces heuristic assumptions with a mathematically consistent Multiple Relaxation Time (MRT) method, which is validated through both analytical and computational benchmarks. Furthermore, the research advances multiphase flow modeling by proposing a new model that integrates the Volume of Fluid (VOF) method with the Partially Saturated Cell (PSC) approach. This integration enables accurate simulations of early-stage shock-droplet interactions, demonstrating both theoretical rigor and practical applicability, as validated by comparisons with published results. The algorithms developed were implemented in parallel using the C++17 standard parallel algo- rithms, providing a hardware-agnostic solution that runs seamlessly on both CPUs and GPUs [65]. The resulting LBM framework delivers a scalable and efficient tool compatible with modern CPU and GPU architectures, significantly extending its applicability to a wider range of high-speed flow simulations. Beyond these immediate contributions, this research lays a solid foundation for future advancements in LBM, including the development of collision models and the investigation of complex physical phe- nomena. These advancements represent a notable step forward in harnessing LBM for state-of-the-art computational fluid dynamics applications.