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.

翻译：本研究探讨了包含曼宁摩擦和非平坦底部地形的源项的浅水方程。我们的主要目标是开发和验证具有双重功能的数值格式：首先，保持模型中的所有稳态；其次，维持解的后期渐近行为，该行为由扩散方程主导，并与长时间和刚性摩擦极限一致。所提出的方法借鉴了《Boscarino等, SIAM Journal on Scientific Computing, 2014》中引入的惩罚技术。通过采用加性隐式-显式龙格-库塔方法，该格式能够确保极限扩散方程的正确渐近行为，同时避免了在扩散极限中常困扰多尺度问题的抛物线型时间步长限制。数值实验展示了所设计格式的高阶精度、渐近保持性和渐近精确性。