This document presents a part of the works done at CELIA Laboratory (CEA, CNRS, Université Bordeaux) in the field of High Energy Density Physics (HEDP). The CELIA plasma physics group is particularly involved in HEDP, Inertial Confinement Fusion (ICF) and Laboratory astrophysics. Flows encountered in HEDP are multi-material and characterized by strong shock waves and large changes in the domain shape due to rarefaction waves. Numerical schemes based on the Lagrangian formalism are good candidates to model this kind of flows since the computational grid follows the fluid motion. This provides accurate results around the shocks as well as a natural tracking of multi-material interfaces. The document is structured as follows. First, we recall the main features of the compatible cell-centered Lagrangian discretization. ALE and ReALE strategy are presented by describing the different phases of our multi-material ALE algorithm. Diffusion scheme used to couple all the physical models is presented with the last extension of the CCLADNS scheme for non-conformal and non-orthogonal meshes in 2D and 3D. Finally CHIC code is introduced and some theoretical studies and experiments are presented.