The Lattice Boltzmann Method is a powerful numerical tool for simulating fluid flows with or without phase change. This poster provides an introduction to the phase-field method and the Lattice Boltzmann Method, demonstrating how the partial differential equations of a coupled hydrodynamics and thermodynamics model can effectively be solved. A phase-field model, derived from a thermodynamic functional based on the grand-potential, is used to simulate the evolution of the phase interface. One of the key advantages of the grand-potential approach is the ability to reformulate the problem in terms of chemical potentials, which are equal in each phase at the thermodynamic equilibrium. Based on that theoretical framework, a showcase of the capabilities of the LBM_Saclay code is presented, ranging from classical 2D two-phase phenomena to more complex 3D three-phase simulations. The code is currently applied for simulating the phase separation of binary or ternary nuclear glasses and the maturation of gels (solid/liquid phase change), involving significant density or viscosity ratio. The simulations were performed on several supercomputers (Jean-Zay and Topaze) equiped with the most recents GPUs.