  We construct a time-asymptotic expansion with pointwise remainder estimates
for solutions to 1D compressible Navier--Stokes equations. The leading-order
term is the well-known diffusion wave and the higher-order terms are newly
introduced family of waves which we call \textit{higher-order diffusion waves}.
In particular, these provide accurate description of the power-law asymptotics
of the solution around the origin $x=0$ where the diffusion wave decays
exponentially. The expansion is valid locally and also globally in the
$L^p(\mathbb{R})$-norm for all $1\leq p\leq \infty$. The proof is based on
pointwise estimates of Green's function.
