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In this study, we investigate the Shallow Water Equations incorporatingsource terms accounting for Manning friction and a non-flat bottom topology.Our primary focus is on developing and validating numerical schemes that servea dual purpose: firstly, preserving all steady states within the model, andsecondly, maintaining the late-time asymptotic behavior of solutions, which isgoverned by a diffusion equation and coincides with a long time and stifffriction limit. Our proposed approach draws inspiration from a penalizationtechnique adopted in {\it{[Boscarino et. al, SIAM Journal on ScientificComputing, 2014]}}. By employing an additive implicit-explicit Runge-Kuttamethod, the scheme can ensure a correct asymptotic behavior for the limitingdiffusion equation, without suffering from a parabolic-type time steprestriction 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 designedschemes.

High order asymptotic preserving and well-balanced schemes for the shallow water equations with source terms