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<p>In this paper, we design a high-order finite difference Weighted Essentially Non-oscillatory (WENO) scheme that preserves the moving water equilibrium for shallow water equations with bottom topography source terms. To achieve the well-balanced property, three key steps are necessary: Firstly, define balance variables, rewrite the governing equations into a’ pre-balanced’ form and rewrite the source term based on moving water steady-state solution information; Secondly, calculate the positive and negative fluxes at integer points using the modified Lax-Friedrichs flux formula; Finally, within the characteristic space, utilize a WENO operator whose weights satisfy scale-invariance and translation-invariance to reconstruct the fluxes at half points. Theoretical verification demonstrates that this scheme can exactly preserve moving water equilibrium. Numerical examples are provided to verify the well-balanced property and good performance of the new scheme.</p>

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A well-balanced Finite Difference WENO Scheme for Shallow Water Equations with Moving water Equilibrium

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Tao Liu,
  • An-qi Deng,
  • Ling-yan Tang

摘要

In this paper, we design a high-order finite difference Weighted Essentially Non-oscillatory (WENO) scheme that preserves the moving water equilibrium for shallow water equations with bottom topography source terms. To achieve the well-balanced property, three key steps are necessary: Firstly, define balance variables, rewrite the governing equations into a’ pre-balanced’ form and rewrite the source term based on moving water steady-state solution information; Secondly, calculate the positive and negative fluxes at integer points using the modified Lax-Friedrichs flux formula; Finally, within the characteristic space, utilize a WENO operator whose weights satisfy scale-invariance and translation-invariance to reconstruct the fluxes at half points. Theoretical verification demonstrates that this scheme can exactly preserve moving water equilibrium. Numerical examples are provided to verify the well-balanced property and good performance of the new scheme.