  In this paper, we examine the stability problem for viscous shock solutions
of the isentropic compressible Navier--Stokes equations, or $p$-system with
real viscosity. We first revisit the work of Matsumura and Nishihara, extending
the known parameter regime for which small-amplitude viscous shocks are
provably spectrally stable by an optimized version of their original argument.
Next, using a novel spectral energy estimate, we show that there are no purely
real unstable eigenvalues for any shock strength.
  By related estimates, we show that unstable eigenvalues are confined to a
bounded region independent of shock strength. Then through an extensive
numerical Evans function study, we show that there is no unstable spectrum in
the entire right-half plane, thus demonstrating numerically that
large-amplitude shocks are spectrally stable up to Mach number $M\approx 3000$
for $1 \le \gamma \leq 3$. This strongly suggests that shocks are stable
independent of amplitude and the adiabatic constant $\gamma$. We complete our
study by showing that finite-difference simulations of perturbed
large-amplitude shocks converge to a translate of the original shock wave, as
expected.
