Asymptotic preserving scheme for the shallow water equations with non-flat bottom topography and Manning friction term
For computational fluid dynamics researchers, this provides a more efficient AP scheme for shallow water flows with friction, though it is an incremental improvement over prior work.
This work extends previous asymptotic preserving (AP) schemes for shallow water equations with bottom topography and Manning friction by removing a penalization term, achieving higher computational efficiency while preserving AP, asymptotically accurate (AA), and well-balanced properties. The new schemes use high-order semi-implicit IMEX-RK time discretization and WENO reconstruction, with implicit treatment of friction being essential for stability.
In our previous work [29], 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 [6]. 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 the high-order WENO reconstruction for first-order derivatives and a central difference scheme 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 (AA) and well-balanced properties, while offering higher computational efficiency compared to our previous schemes in [29], 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.