Asymptotic Preserving Semi-Implicit Scheme for the Shallow Water Equations with Non-Flat Bottom Topography and Manning Friction Term

In [G. Huang, S. Boscarino and T. Xiong, High order asymptotic preserving and well-balanced schemes for the shallow water equations with source terms, Commun. Comput. Phys. 35 2024, 5, 1229-1262], we proposed a class of high-order asymptotic preserving (AP) finite difference weighted essentially non-oscillatory (WENO) schemes for solving the shallow water equations (SWEs) with bottom topography and Manning friction, utilizing a penalization technique inspired by [S. Boscarino, P. G. LeFloch and G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput. 36 2014, 2, A377-A395]. Although the added weighted diffusive term enhanced stability, it increased computational cost and slowed down the convergence rate in the intermediate regime between convection and diffusion. In this paper, we extend our previous study by removing the penalization while preserving the AP property. To achieve this, we employ a high order semi-implicit implicit-explicit Runge-Kutta (SI-IMEX-RK) time discretization, coupled with high-order WENO reconstructions for first-order derivatives and central difference schemes for second-order spatial derivatives. This combination yields a class of fully high-order schemes. Theoretical analysis and numerical experiments demonstrate that the proposed schemes retain AP, asymptotically accurate and well-balanced properties, while offering higher computational efficiency compared to our previous scheme in Huang, Boscarino and Xiong (2024), especially in the intermediate regime between convection and diffusion. Moreover, treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise, the scheme fails to converge to the limiting equations. This indicates that implicit treatment of Manning friction is necessary for the stability of the method.