Shock structure in viscous fluids

In the context of shock-capturing numerical methods for Lagrangian gas dynamics, shock wave resolution and localization remain important practical problems. Any improvement in our understanding of 2D and 3D problems must come from a greater understanding of corresponding 1D problems. Explicit modeling of the transition profile or structure of 1D viscous shock waves goes back to the work of Becker (1921) and many others. For steady shock waves, the dependence of several shape parameters defining the structure of the smeared-transition profile, such as front location and width, have been studied previously for various forms of the viscous term including the well-known Richtmyer-von Neumann viscosity model. In this presentation, we compare several approximate shock profiles obtained using a Lagrangian, staggered-grid finite difference scheme against exact solutions of their PDE-based model counterparts using the aforementioned shape parameters to quantify the effects of different models for artificial viscosity. One objective of this comparison is to develop a structure-preserving, artificial viscosity optimization procedure that can be built into existing Arbitrary Lagrangian-Eulerian methods.