Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving property, adopt similar idea of WENO-XS (Xing and Shu (2005) [30]) to balance flux gradients source terms. The fluxes original are reconstructed by nonlinear HWENO reconstructions while other derivative approximated high-degree polynomials directly. And an limiter is applied derivatives equilibrium variables time discretization step control spurious oscillations which maintains property. Instead using five-point stencil scheme, proposed only needs compact three-point reconstruction. Various benchmark examples one two dimensions presented show accuracy, preserves steady-state solution, has better resolution, more accurate efficient, essentially non-oscillatory.