Explore high-order finite element methods for shock hydrodynamics simulations in this seminar presented by LLNL computational mathematician Vladimir Tomov. Delve into Arbitrary Lagrangian-Eulerian (ALE) formulation, covering Lagrangian shock hydrodynamics on curved meshes, multi-material closure models, and coupling to multigroup radiation diffusion. Examine mesh optimization techniques, including r-adaptivity and surface fitting of high-order meshes, as well as advection-based remap with nonlinear sharpening of material interfaces. Discover computationally efficient finite element kernels based on partial assembly and sum factorization. Gain insights into existing methods, outstanding research challenges, and ongoing work in the field of ALE hydrodynamics.