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

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving convection–diffusion equations on non-smooth Cartesian meshes. WENO (weighted essentially non-oscillatory) integration and linear interpolation for the derivatives are used to compute the numerical fluxes based on the point values of the solution. The objective is to obtain a high order scheme which, for two space dimension, has a computational cost 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 it does not have the restriction on mesh smoothness of the traditional high order conservative finite difference schemes, hence it would be more flexible for the resolution of sharp layers. The principles of residual distribution schemes are adopted to obtain steady state solutions. The distribution of residuals resulted from the convective and diffusive parts of the PDE is carefully designed to maintain the high order accuracy. The proof of a Lax-Wendroff type theorem is provided for convergence towards weak solutions in one and two dimensions under additional assumptions. Extensive numerical experiments for one and two-dimensional scalar problems and systems confirm the high order accuracy and good quality of our scheme to resolve the inner or boundary layers.