High Order Asymptotic Preserving and Well-Balanced Schemes for the Shallow Water Equations with Source Terms

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, maintain- ing 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 [Boscarino et al., SIAM Journal on Scientific Computing, 2014]. By employing an addi- tive 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. Nu- merical experiments are performed to illustrate high order accuracy, asymptotic pre- serving, and asymptotically accurate properties of the designed schemes.