A Node-Centered Artificial Viscosity Method for Two-Dimensional Lagrangian Hydrodynamics Calculations on a Staggered Grid

This work deals with the simulation of two-dimensional Lagrangian hydrodynamics problems. Our objective is the development of an artificial viscosity that is to be used in conjunction with a staggered placement of variables: thermodynamics variables are centered within cells and position and fluid velocity at vertices. In [J. Comput. Phys., 228 (2009), 2391-2425], Maire develops a high-order cell-centered scheme for solving the gas dynamics equations. The numerical results show the accuracy and the robustness of the method, and the fact that very few Hourglass-type deformations are present. Our objective is to establish the link between the scheme of Maire and the introduction of artificial viscosity in a Lagrangian code based on a staggered grid. Our idea is to add an extra degree of freedom to the numerical scheme, which is an approximation of the fluid velocity within cells. Doing that, we can locally come down to a cell-centered approximation and define the Riemann problem associated to discrete variable discontinuities in a very natural way. This results in a node-centered artificial viscosity formulation. Numerical experiments show the robustness and the accuracy of the method, which is very easy to implement.