We present a comparative study of two different Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials while the density is advanced by a modal density evolution, which has not been extended to RZ coordinates. Two different approaches are used to discretize the evolution equations ,i.e., the true volume approach and the area-weighted approach. For the true volume approach, the DG equations are derived using on the true 3D volume that is consistent with the geometry conservation law (GCL). A multidirectional approximate Riemann problem at the element surface nodes can be solved using the surface areas in axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based in the 2D Cartesian geometry that is rotated about the axis of symmetry using an element average radius. With this approach, a multidirectional approximate Riemann problem identical to the one in 2D Cartesian geometry can be solved. This area-weighted approach conserves mass, and in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. A suite of test problems are calculated to demonstrate stable mesh motion and the expected second order accuracy of these methods.
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