A fourth-order Lagrangian discontinuous Galerkin method using a hierarchical orthogonal basis on curvilinear grids

The existing high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic methods are restricted to using quadratic meshes with quadratic polynomials (P2), which in turn, yield up to third-order accuracy. These existing DG hydrodynamic schemes, when extended to work with cubic meshes and cubic polynomials (P3), can be unstable on strong-shock problems. Therefore, this paper presents a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate compressible material dynamics ( e.g., gasses, fluids, and solids) with strong-shocks using cubic meshes and cubic polynomials, and delivers up to fourth-order accuracy on smooth flows. The stability on shock problems is achieved using new hierarchical orthogonal basis functions and a new subcell mesh stabilization (SMS) scheme for cubic meshes. 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 curvilinear meshes.