Implicit Wall‐Constrained Matrix Reconstruction for Finite Difference Boundary Treatment on Curved Walls

ABSTRACT

When solving partial differential equations using finite‐difference methods, near‐boundary accuracy is often limited by truncation of finite‐difference stencils and limitations of boundary treatments. These limitations often lead to local order reduction, making it challenging to maintain globally consistent second‐order accuracy. To address these challenges, this paper proposed a constrained matrix reconstruction optimization (CMRO) method with an implicit wall constraint. The method employs a matrix‐based simultaneous solution strategy that integrates physical boundary constraints and interior discrete equations into a unified coupled system while incorporating a ghost‐point technique to preserve stencil compactness. Results indicate that, within a non‐iterative framework, the proposed method helps maintain second‐order accuracy near boundaries in the tested cases. Timing analysis shows that the additional cost is confined to wall nodes and remains modest relative to the overall time‐marching cost. Numerical examples demonstrate the method's accuracy consistency, robustness, and bounded computational overhead, offering a practical approach for high‐precision simulations with complex boundaries and demonstrating its value for engineering applications.