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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 deformationgradient fields within a cell, are represented with a polynomial constructedfrom a novel hierarchical orthogonal basis about the center of mass, whichdecouples the moments of the solution because the mass matrix is diagonal. Thediscontinuity in the polynomials at the cell boundary is addressed by solving amulti-directional Riemann problem at the vertices of the cell and a 1D Riemannproblem at additional non-vertex quadrature points along the edges so that thesurface integral is exact for the polynomial order. The uniqueness lies in thatthe vertices of the curvilinear grid work as the quadrature points for thesurface integral of DG methods. To ensure robust mesh motion, the pressure forthe Riemann problem accounts for the difference between the density variationover the cell and a density field from subcell mesh stabilization (SMS). Theaccuracy and robustness of the new high-order accurate Lagrangian DGhydrodynamic method is demonstrated by simulating a diverse suite ofchallenging test problems covering gas and solid dynamic problems on curvedgrids.

A discontinuous Galerkin method based on a hierarchical orthogonal basis for Lagrangian hydrodynamics on curvilinear grids