-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 7: 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 only to the gradient of T$_{yy}$ . The shear viscosity, heat conductivity, heat capacity, and thermal relaxation times are all taken equal to unity. Q$_{x}$ = − κ ( ∇ T$_{xx}$ + ∇ T$_{yy}$ ) / 2 = − ( ∇ T$_{xx}$ + ∇ T$_{yy}$ ) / 2 [Choice 2] ; It is good fortune that the shockwave equations we summarize here are relatively easy to solve numerically. The usual numerical method is the “backward Euler” scheme $^{2}$. One starts near the “hot” boundary and integrates backward, using a first-order difference scheme. That approach fails here, due to the temperature relaxation terms, which are ex- ponentially unstable in the time-reversed case. An integration forward in time is required in the presence of relaxation. A successful “staggered-grid” (two separate spatial grids) algorithm results if the density ρ$_{c}$ is defined at cell centers and energy, temperature, and pressure are defined at the nodes which bound the cells $^{18,19}$. This algorithm follows the dynamics correctly and converges nicely to the stationary profiles shown in Figures 7 and 8. A computational mesh spacing of dx = 0 . 1 is sufficient, using the second-order spatial differencing scheme outlined in References 18 and 19 with fourth-order Runge-Kutta time integration. In the early days of shockwave modeling this computational simplicity was by no means apparent, so that there is an abundant literature on the stability of numerical methods for the shockwave problem $^{2}$. Now, in the early days of tensor-temperature models, the main challenge is to develop well-posed constitutive equations consistent with both the conservation laws and the empirical results from molecular dynamics. Interesting aspects of both solutions are (i) the minimum in P$_{yy}$ ( x ), which suggests the need for bulk viscosity in modeling molecular dynamics results, and (ii) the pronounced