A nonlinear immersed boundary method for weighted compact nonlinear schemes

Weighted compact nonlinear schemes are a class of high-order finite difference schemes that are widely used in applications. The schemes are flexible in the choice of numerical fluxes. When applied to complex configurations, curvilinear grids are often applied, where the symmetric conservative metric method can be used to ensure geometric conservation laws. However, for complex configurations it may be difficult to generate high quality curvilinear grids. Thus, we confine the study in this paper to Cartesian grids and develop a nonlinear immersed boundary method to deal with the boundary. The developed method is applicable to different kinds of boundary conditions. In addition, compared with the traditional immersed boundary method, this new method can handle problems with shocks near boundary. Both one- and two-dimensional cases are studied into details, with corresponding numerical results showing the validity of the proposed method.