A Hybrid WENO Method for Hyperbolic Conservation Laws on the Polygonal Mesh With Mixed Elements

ABSTRACT In this paper, we present one class of high order hybrid weighted essentially non‐oscillatory (WENO) schemes for solving the hyperbolic conservation laws on polygonal meshes based on the finite volume framework. The method is designed for unstructured polygon meshes consisting of both triangular and quadrilateral elements. Two key features characterize our proposed approach. We first use a distance‐dependent weighted least squares reconstruction method, where the distances between stencil cells and the target cell are incorporated into the polynomial reconstruction to determine the reconstruction coefficients. Then, a hybrid strategy is introduced that uses an efficient troubled cell indicator to selectively apply characteristic decomposition and nonlinear weights, thereby reducing computational cost while preserving accuracy. The proposed scheme effectively suppresses spurious oscillations near discontinuities, maintains high resolution, and improves computational efficiency. Numerical experiments on mixed element meshes demonstrate that the hybrid method accurately captures discontinuities, produces high‐resolution solutions, and achieves a notable reduction in computational expense.