Using a curvilinear grid to construct symmetry-preserving discretizations for Lagrangian gas dynamics

The goal of this paper is to construct discretizations for the equations of Lagrangian gas dynamics that preserve plane, cylindrical, and spherical sy mmetry in the solution of the original differential equations. The new meth od uses a curvilinear grid that is reconstructed from a given logically rec tangular distribution of nodes. The sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles. Our proce dure is exact for straight lines and circles; that is, it reproduces rectan gular and polar grids exactly. We use the method of support operators to co nstruct a conservative finite-difference method that we demonstrate will pr eserve spatial symmetries for certain choices of the initial grid. We also introduce a "curvilinear" version of artificial edge viscosity that also pr eserves symmetry. We present numerical examples to demonstrate our theoreti cal considerations and the robustness of the new method.