  We present a new high-order accurate Lagrangian discontinuous Galerkin (DG)
hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids,
and solids) with up to fourth-order accuracy on cubic meshes. The variables,
such as specific volume, velocity, specific total energy, and deformation
gradient fields within a cell, are represented with a polynomial constructed
from a novel hierarchical orthogonal basis about the center of mass, which
decouples the moments of the solution because the mass matrix is diagonal. The
discontinuity in the polynomials at the cell boundary is addressed by solving a
multi-directional Riemann problem at the vertices of the cell and a 1D Riemann
problem at additional non-vertex quadrature points along the edges so that the
surface integral is exact for the polynomial order. The uniqueness lies in that
the vertices of the curvilinear grid work as the quadrature points for the
surface integral of DG methods. To ensure robust mesh motion, the pressure for
the Riemann problem accounts for the difference between the density variation
over the cell and a density field from subcell mesh stabilization (SMS). The
accuracy and robustness of the new high-order accurate Lagrangian DG
hydrodynamic method is demonstrated by simulating a diverse suite of
challenging test problems covering gas and solid dynamic problems on curved
grids.
