  In this paper, we develop a high order residual distribution (RD) method for
solving steady state conservation laws in a novel Hermite weighted essentially
non-oscillatory (HWENO) framework recently developed in [23]. In particular, we
design a high order HWENO reconstructions for the integrals of source term and
fluxes based on the point values of the solution and its spatial derivatives,
and the principles of residual distribution schemes are adapted to obtain
steady state solutions. The proposed novel HWENO framework enjoys two
advantages. First, compared with the traditional HWENO framework, the proposed
methods do not need to introduce additional auxiliary equations to update the
derivatives of the unknown function, and compute them from the current value
and the old spatial derivatives. This approach saves the computational storage
and CPU time, which greatly improves the computational efficiency of the
traditional HWENO framework. Second, compared with the traditional WENO method,
reconstruction stencil of the HWENO methods becomes more compact, their
boundary treatment is simpler, and the numerical errors are smaller at the same
grid. Thus, it is also a compact scheme when we design the higher order
accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical
experiments for one and two-dimensional scalar and systems problems confirm the
high order accuracy and good quality of our scheme.
