High Order Residual Distribution Conservative Finite Difference HWENO Scheme for Steady State Problems

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.