High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving steady state hyperbolic conservation laws on non-smooth Cartesian or other structured curvilinear meshes. WENO (weighted essentially non-oscillatory) integration is used to compute the numerical fluxes based on the point values of the solution, and the principles of residual distribution schemes are adapted to obtain steady state solutions. In two space dimension, the computational cost of our scheme is comparable to that of a high order WENO finite difference scheme and is therefore much lower than that of a high order WENO finite volume scheme, yet the new scheme does not have the restriction on mesh smoothness of the traditional high order conservative finite difference schemes. A Lax–Wendroff type theorem is proved for convergence towards weak solutions in one and two dimensions, and extensive numerical experiments are performed for one- and two-dimensional scalar problems and systems to demonstrate the quality of the new scheme, including high order accuracy on non-smooth meshes, conservation, and non-oscillatory properties for solutions with shocks and other discontinuities.