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

This thesis contains two related topics, both arising from the numerical approximation for steady state problems.;In the first part, we develop 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. A Lax-Wendroff type theorem is proved for convergence towards weak solutions in one and two dimensions. Numerical experiments are performed for one and two dimensional scalar problems and systems to demonstrate the quality of the scheme, including high order accuracy on non-smooth meshes, conservation, and non-oscillatory properties for solutions with shocks and other discontinuities.;In the second part, we generalize the scheme proposed in the first part to convection-diffusion equations. WENO integration and linear interpolation for the derivatives are used to compute the numerical fluxes based on the point values of the solution. The distribution of residuals resulted from 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 with 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.