  In this study, we investigate the Shallow Water Equations incorporating
source terms accounting for Manning friction and a non-flat bottom topology.
Our primary focus is on developing and validating numerical schemes that serve
a dual purpose: firstly, preserving all steady states within the model, and
secondly, maintaining the late-time asymptotic behavior of solutions, which is
governed by a diffusion equation and coincides with a long time and stiff
friction limit. Our proposed approach draws inspiration from a penalization
technique adopted in {\it{[Boscarino et. al, SIAM Journal on Scientific
Computing, 2014]}}. By employing an additive implicit-explicit Runge-Kutta
method, the scheme can ensure a correct asymptotic behavior for the limiting
diffusion equation, without suffering from a parabolic-type time step
restriction which often afflicts multiscale problems in the diffusive limit.
Numerical experiments are performed to illustrate high order accuracy,
asymptotic preserving, and asymptotically accurate properties of the designed
schemes.
