High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations

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.