  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 the well-balance
property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput.
Phys., 208 (2005), 206-227.] to balance the flux gradients and the source
terms. The fluxes in the original equation are reconstructed by the nonlinear
HWENO reconstructions while other fluxes in the derivative equations are
approximated by the high-degree polynomials directly. And an HWENO limiter is
applied for the derivatives of equilibrium variables in time discretization
step to control spurious oscillations which maintains the well-balance
property. Instead of using a five-point stencil in the same fifth-order WENO-XS
scheme, the proposed HWENO scheme only needs a compact three-point stencil in
the reconstruction. Various benchmark examples in one and two dimensions are
presented to show the HWENO scheme is fifth-order accuracy, preserves
steady-state solution, has better resolution, is more accurate and efficient,
and is essentially non-oscillatory.
