Boundary treatments across block interfaces for high-order cell-centered finite difference method with shocks and distortion

Constructing boundary schemes within finite difference methods that simultaneously provide numerical stability, robust shock capturing, high-order accuracy, and global integral conservation remains a formidable challenge. In this paper, based on the weighted compact nonlinear scheme (WCNS) framework, we devise a multi-block boundary treatment for block interfaces with grid distortion and possible shocks that (i) retains high order of discretization accuracy in smooth flows, (ii) maintains strong robustness across shock discontinuities, and (iii) rigorously enforces global integral conservation throughout the simulation. A series of strategies has been applied. First, we begin with the multi-resolution WCNS, which blends 1-, 3-, and 5-point stencils for shock capture. By redesigning the nonlinear weights, the scheme now regains full third-order accuracy whenever a discontinuity contaminates only the 5-point stencil, whereas the original formulation collapsed to first order in that situation. Second, exploiting the compact stencils near boundaries, we devise four nearly one-sided nonlinear interpolation schemes without directly using cross-interface data. The resulting boundary scheme is high-order accurate in smooth, distorted grids and stays sharp and non-oscillatory when a shock lies on the interface. Finally, to achieve global integral conservation of the conserved variables, we propose a hybrid conservative difference scheme that combines face-to-cell and cell-to-cell stencils, driving conservation errors to machine precision. At last, the proposed schemes are validated through a series of numerical tests. The results demonstrate that even with grid distortion across block interfaces, the proposed scheme ensures the desired order of accuracy in smooth flow and avoids oscillations near shock locations.