Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations

A new SUPG-stabilized formulation for Lagrangian Hydrodynamics of materials satisfying the Mie–Grüneisen equation of state was presented in the first paper of the series [G. Scovazzi, M.A. Christon, T.J.R. Hughes, J.N. Shadid, Stabilized shock hydrodynamics: I. A Lagrangian method, Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2006.08.008]. The present article investigates in more detail the design of the SUPG stabilization operator, focusing on its multiscale and physical interpretations. Connections with the analysis of Kuropatenko [V.F. Kuropatenko, On difference methods for the equations of hydrodynamics, in: N.N. Janenko (Ed.), Difference Methods for Solutions of Problems of Mathematical Physics, I, American Mathematical Society, Providence, RI, 1967] for discontinuity-capturing operators in the limit of weak shocks are shown. Galilean invariance requirements for the SUPG operator are explored and corroborated by numerical evidence. This work is intended to elucidate the profound physical significance of the SUPG operator as a subgrid interaction model.