arXiv subjects

Guanlan Huang

Publications and source records attributed to Guanlan Huang.

4 recordsLinked to original sources

arXiv2026

Asymptotic preserving scheme for the shallow water equations with non-flat bottom topography and Manning friction term

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.

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arXiv2023

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, 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.

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arXiv2022

High order asymptotic preserving well-balanced finite difference WENO schemes for all Mach full Euler equations with gravity

In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of potential temperature, we design a novel well-balanced finite difference WENO scheme for the conservative variables, which can be proven to be both asymptotic preserving and asymptotically accurate in the incompressible limit. Extensive numerical experiments were provided to validate these properties.

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arXiv2021

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

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