Triangular meshes are appealing for meshing complex geometries; however, linear triangles can perform poorly on large deformation problems with purely Lagrangian methods due to unphysical locking. The limitations with linear triangular elements can be overcome by using higher-order triangular elements that have edges that can bend. We present a Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured meshes with quadratic triangular elements. The specific volume, velocity, and specific total energy fields are approximated with linear Taylor expansions. The velocity at the element surface nodes, and the corresponding forces, are calculated by solving a multidirectional approximate Riemann problem. Two different approximate Riemann solvers are studied. An explicit TVD Runge-Kutta method is used to temporally advance the solution. The Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Test problem results are presented to demonstrate the robustness and convergence order of the method. Results are presented for both linear and quadratic triangular elements to demonstrate the merits of using meshes with quadratic triangular elements.
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