A high-order interpolation method for hanging points based on the adaptive Cartesian mesh

The adaptive mesh refinement technology is widely employed in Cartesian grid methods to improve computational efficiency by refining the mesh only in regions requiring high resolution. However, local refinement introduces hanging points at coarse–fine interfaces, making the construction of accurate and robust interpolation methods a key challenge for high-order numerical discretization. In this work, a third-order hanging-point interpolation framework is developed for both first- and second-type hanging points. For different types of hanging points, we propose (1) the least squares interpolation and (2) multi-stencil interpolation weighted by weighted essentially non-oscillatory and exponential inverse-distance strategies. Numerical experiments demonstrate that the proposed method achieves third-order accuracy in smooth regions and maintains good robustness in the presence of shocks and shock interactions. In addition, the proposed interpolation framework is independent of the subsequent reconstruction procedure and can be readily integrated with different schemes for adaptive Cartesian methods.