  In this paper we analyze the large-time behavior of the augmented Burgers
equation. We first study the well-posedness of the Cauchy problem and obtain
$L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by
showing that the influence of the convolution term $K*u_{xx}$ is the same as
$u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme
that preserves this asymptotic behavior, by introducing two correcting factors
in the discretization of the non-local term. Numerical experiments illustrating
the accuracy of the results of the paper are also presented.
