  We have considered a shock wave as a surface of discontinuity and computed
the entropy production using non-equilibrium thermodynamics for surfaces. The
results from this method, which we call the "Gibbs excess method" (GEM), were
compared with results from three alternative methods, all based on the entropy
balance in the shock front region, but with different assumptions about local
equilibrium. Non-equilibrium molecular dynamics (NEMD) simulations were used to
simulate a thermal blast in a one-component gas consisting of particles
interacting with the Lennard-Jones/spline potential. This provided data for the
theoretical analysis. Two cases were studied, a weak shock with Mach number $M
\approx 2$ and a strong shock with $M \approx 6$ and with a Prandtl number of
the gas $Pr \approx 1.4$ in both cases. The four theoretical methods gave
consistent results for the time-dependent surface excess entropy production for
both Mach numbers. The internal energy was found to deviate only slightly from
equilibrium values in the shock front. The pressure profile was found to be
consistent with the Navier-Stokes equations. The entropy production in the weak
and strong shocks were approximately proportional to the square of the Mach
number and decayed with time at approximately the same relative rate. In both
cases, some 97 \% of the total entropy production in the gas occurred in the
shock wave. The GEM showed that most of the shock's kinetic energy was
converted reversibly into enthalpy and entropy, and a small amount was
dissipated as produced entropy. The shock waves traveled at almost constant
speed and we found that the overpressure determined from NEMD simulations
agreed well with the Rankine-Hugoniot conditions for steady-state shocks.
