  In this paper, a compact and high order ADER (Arbitrary high order using
DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for
hyperbolic conservation laws. The newly-developed method employs the
Lax-Wendroff procedure to convert time derivatives to spatial derivatives,
which provides the time evolution of the variables at the cell interfaces. This
information is required for the simple HWENO reconstructions, which take
advantages of the simple WENO and the classic HWENO. Compared with the original
Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages.
Firstly, RK-HWENO method must solve the additional equations for
reconstructions, which is avoided for the new method. Secondly, the SHWENO
reconstruction is performed once with one stencil and is different from the
classic HWENO methods, in which both the function and its derivative values are
reconstructed with two different stencils, respectively. Thus the new method is
more efficient than the RK-HWENO method. Moreover, the new method is more
compact than the existing ADER-WENO method. Besides, the new method makes the
best use of the information in the ADER method. Thus, the time evolution of the
cell averages of the derivatives is simpler than that developed in the work [Li
et. al., 447 (2021), 110661.]. Numerical tests indicate that the new method can
achieve high order for smooth solutions both in space and time, keep
non-oscillatory at discontinuities.
