Abstract A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the source term in the finite difference scheme [Xing and Shu, J. Comput. Phys. 208 (2005)] and the reconstruction technique in the finite volume WENO scheme [Xing and Shu, J. Comput. Phys. 214 (2006)] are adopted. The proposed scheme can be proved mathematically to maintain the exact C-property and demonstrates numerically that it is well-balanced by construction for the stationary water surface. Moreover, the local characteristic projections are employed to further mitigate the Gibbs oscillations. The proposed generic high order WENO schemes not only achieve high order accuracy but also capture the high gradients/shock waves essentially non-oscillatory. Meanwhile, the small perturbation problems can be resolved well on a coarse grid.

中文翻译：

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浅水方程的高阶均衡有限差分 WENO 基于插值的方案

摘要 针对浅水方程，提出了基于高阶均衡有限差分加权基本非振荡(WENO)插值方案的广义形式的数值框架。它展示了比经典的基于 WENO 重建的方案更灵活的构造过程。加权紧非线性方案和有限差分替代 WENO 方案是两种特殊情况。为了保持精确的 C 属性，有限差分方案中源项的分裂技术 [Xing and Shu, J. Comput. 物理。208 (2005)] 和有限体积 WENO 方案中的重建技术 [Xing and Shu, J. Comput. 物理。214 (2006)] 通过。所提出的方案可以在数学上证明以保持精确的 C 性质，并在数值上证明它通过构造固定水面是很好的平衡。此外，采用局部特征投影来进一步减轻吉布斯振荡。所提出的通用高阶 WENO 方案不仅实现了高阶精度，而且还捕获了基本上非振荡的高梯度/冲击波。同时，在粗网格上可以很好地解决小扰动问题。