0.0 0.1 0.2 0.3 -3.5 < x < +3.5 T van der Waals Shock 0.0 1.0 2.0 3.0 -3.5 < x < +3.5 P > P xx yy e ρ Figure 6: Stationary spatial profile for a one-dimensional shockwave according to the usual Navier-Stokes-Fourier equations for the model fluid: P$_{eq}$ = ρe ; e = ( ρ/ 2) + kT with unit shear viscosity, zero bulk viscosity, and unit Fourier heat conductivity. Here the temperature T (left) is a scalar, as in conventional continuum mechanics. the molecular dynamics profiles shown in Figures 3, with space as the abscissa. Exactly the same profiles can alternatively be expressed with time as the abscissa, as in Figure 5. To change from space-based to time-based profiles requires use of the ratio ( dx/dt ) ≡ u : ∫ t 0 dt ′ = ∫ x x 0 dx $^{′}$/u ( x $^{′}$) ; t = 0 ↔ x = x$_{0}$ . where u ( x ) is the hydrodynamic flow velocity. Thus all the spatial snapshots or equivalent temporal wave profiles catalog the sequence of time-ordered states through which the particles in a typical volume (initially at x$_{0}$ ) pass as they transit the shockwave. Because the conventional Navier-Stokes-Fourier approach, illustrated in Figure 6, as- sumes a scalar temperature, T = T$_{xx}$ = T$_{yy}$ , several modifications of the continuum de- scription need to be made to model the two-temperature results of Figures 3-5 found with molecular dynamics, with T$_{xx}$ = T$_{yy}$ . In Section III we describe simple modifications of the Navier-Stokes-Fourier constitutive and flow equations, along with a numerical method which converges nicely to give stationary shockwave profiles in the two-temperature case. Section IV is reserved for a summary and our concluding remarks, including sugges- tionns for adapting our ideas to detailed two- and three-dimensional descriptions of the fluctuations in nonequilibrium systems.