  In this paper, a simple fifth-order finite difference Hermite WENO (HWENO)
scheme combined with limiter is proposed for one- and two- dimensional
hyperbolic conservation laws. The fluxes in the governing equation are
approximated by the nonlinear HWENO reconstruction which is the combination of
a quintic polynomial with two quadratic polynomials, where the linear weights
can be artificial positive numbers only if the sum equals one. And other fluxes
in the derivative equations are approximated by high-degree polynomials
directly. For the purpose of controlling spurious oscillations, an HWENO
limiter is applied to modify the derivatives. Instead of using the modified
derivatives both in fluxes reconstruction and time discretization as in the
modified HWENO scheme (J. Sci. Comput., 85:29, 2020), we only apply the
modified derivatives in time discretization while remaining the original
derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme,
the proposed HWENO scheme is simpler, more accurate, efficient and higher
resolution. In addition, the HWENO scheme has a more compact spatial
reconstructed stencil and greater efficiency than the classical fifth-order
finite difference WENO scheme of Jiang and Shu. Various benchmark numerical
examples are presented to show the fifth-order accuracy, great efficiency, high
resolution and robustness of the proposed HWENO scheme.
