A modern concept of Lagrangian hydrodynamics

We give a modern overview of Lagrangian hydrodynamics as it is implemented in Lagrangian codes for modeling compressible fluid dynamics. The main result is to show that artificial viscosity, often treated as a numerical knob to control unphysical oscillations near shocks, actually represents a physical process and it is required to produce accurate simulation results for any compressible flow. In the first part, we review the origins of two numerical tools, artificial viscosity and finite volume methods. Then, we mathematically derive a set of partial differential equations (PDE) with a high-order finite volume approximation that contains a new length scale, the observer, as part of the equations that arises representing the discretization. Associated with that length scale, there are new inviscid fluxes which are the artificial viscosity that was first formulated by Richtmyer and von Neumann. Additionally an artificial heat flux postulated by Noh appears that is not typically included in Lagrangian codes. The results are discussed in the context of bi–velocity hydrodynamics. Finally, the conclusion introduces speculations about the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.