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

  • Guanlan Huang , Sebastiano Boscarino and Tao Xiong EMAIL logo
Published/Copyright: April 30, 2025

Abstract

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.

MSC 2020: 35L65; 65L04; 65M06

Award Identifier / Grant number: 2022YFA1004500

Award Identifier / Grant number: 92270112

Award Identifier / Grant number: 2023J02003

Funding statement: The first and third authors are partially supported by National Key R & D Program of China No. 2022YFA1004500, NSFC grant No. 92270112, NSF of Fujian Province No. 2023J02003. Sebastiano Boscarino is supported for this work by (1) the Spoke 1 “FutureHPC & BigData” of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4 Componente 2 Investimento 1.4: Potenziamento strutture di ricerca e creazione di “campioni nazionali di R & S (M4C2-19 )”; by (2) the Italian Ministry of Instruction, University and Research (MIUR) to support this research with funds coming from PRIN Project 2022 (2022KA3JBA), entitled “Advanced numerical methods for time dependent parametric partial differential equations and applications”; (3) from Italian Ministerial grant PRIN 2022 PNRR “FIN4GEO: Forward and Inverse Numerical Modeling of hydrothermal systems in volcanic regions with application to geothermal energy exploitation.”, (No. P2022BNB97). S. Boscarino is a member of the INdAM Research group GNCS.

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Received: 2024-11-25
Revised: 2025-03-23
Accepted: 2025-03-26
Published Online: 2025-04-30
Published in Print: 2025-07-01

© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston

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