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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 symmetry in the solution of the original differential equations. The new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes. The sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles. Our procedure is exact for straight lines and circles; that is, it reproduces rectangular and polar grids exactly. We use the method of support operators to construct a conservative finite-difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid. We also introduce a “curvilinear” version of artificial edge viscosity that also preserves symmetry. We present numerical examples to demonstrate our theoretical considerations and the robustness of the new method.