-0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 -5.0 < x < +5.0 Temperatures Heat Flux 0.0 1.0 2.0 3.0 -5.0 < x < +5.0 xx and yy Pressures u e ρ Figure 8: Typical solution of the generalized Navier-Stokes-Fourier equations for the van der Waals model with both heat and work contributing to T$_{xx}$ and with the heat flux responding equally to the gradients of both T$_{xx}$ and T$_{yy}$ . The shear viscosity, heat conductivity, heat capacity, and thermal relaxation times are all taken equal to unity. maximum in T$_{xx}$ ( x ), leading the response of T$_{yy}$ and roughly equal in magnitude to that found in the dynamical results of Section II. The physical ideas incorporated in this simplest approach are four: (i) the pressure and the work done can usefully be separated into a “cold” part and a “thermal” part; (ii) the heat flux Q responds to a linear combination of the temperature gradients ∇ T$_{xx}$ and ∇ T$_{yy}$ in the usual way, supplemented by (iii) the thermal relaxation of the thermal anisotropicity, and (iv) separate linear combinations of the work done and heat absorbed contribute to T$_{xx}$ and T$_{yy}$ throughout the shock compression process. Here the total pressure, P = P Φ + P $^{K}$, contains potential and kinetic components, measurable separately with molecular dynamics. These extensions of the Navier-Stokes approach closely parallel the relaxation-time treatments of strong ideal-gas shockwaves carried out by Xu, Josyula, Holian, and Mareschal $^{11,14}$. Our more general approach necessarily differs from theirs by allowing for contributions from the potential energy to temperature changes and the transfer of heat. The pressure profiles shown in Figures 7 and 8 also indicate the need for bulk viscosity, in that the molecular dynamics results show a monotone-increasing P$_{yy}$ , in contrast to the distinct minimum found here in the absence of bulk viscosity. We turn next to a slightly more sophisticated model, an extension of Gr¨ uneisen’s equilibrium equation of state.