The discretization of the Euler equations of gas dynamics (“compressible hydrodynamics”) in a moving material frame is at the heart of many multi-physics simulation codes. The Arbitrary Lagrangian-Eulerian (ALE) framework is frequently applied in these settings in the form of a Lagrange phase, where the hydrodynamics equations are solved on a moving mesh, followed by a three-part “advection phase” involving mesh optimization, field remap and multi-material zone treatment.
This talk presents a general Lagrangian framework  for discretization of compressible shock hydrodynamics using high-order finite elements. The use of high-order polynomial spaces to define both the mapping and the reference basis functions in the Lagrange phase leads to improved robustness and symmetry preservation properties, better representation of the mesh curvature that naturally develops with the material motion and significant reduction in mesh imprinting. We will discuss the application of the curvilinear technology to the “advection phase” of ALE, including a DG-advection approach for conservative and monotonic high-order finite element interpolation (remap), as well as to coupled physics, such as electromagnetic diffusion. We will also review progress in robust and efficient algorithms for high-order mesh optimization, matrix-free preconditioning, high-order time integration and matrix-free monotonicity, which are critical components for the successful use of high-order methods in the compressible ALE settings
In addition to their mathematical benefits, high-order finite element discretizations are a natural fit for modern HPC hardware, because their order can be used to tune the performance, by increasing the FLOPs/bytes ratio, or to adjust the algorithm for different hardware. In this direction, we will present some of our work on scalable high-order finite element software that combines the modular finite element library MFEM  and the high-order shock hydrodynamics code BLAST , where we will demonstrate the benefits of our approach with respect to strong scaling and GPU acceleration. Finally, we will give a brief update on related efforts in the co-design Center for Efficient Exascale Discretizations (CEED) in the Exascale Computing Project (ECP) of the DOE .
 “High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics”, V. Dobrev and Tz. Kolev and R. Rieben, SIAM Journal on Scientific Computing, (34) 2012, pp.B606-B641.
 MFEM: Modular finite element library, http://mfem.org.
 BLAST: High-order shock hydrodynamics, http://llnl.gov/casc/blast.
 Center for Efficient Exascale Discretizations, http://ceed.exascaleproject.org.