Numerical schemes critically affect the accuracy of computational fluid dynamics. This study presents an improved Harten–Lax–Leer–Einfeldt Plus (HLLE++) scheme tailored for cell-centered finite volume methods (CCFVMs) on unstructured hybrid meshes. The HLLE++ scheme, originally designed to address challenges in wide-speed-range flow simulations (e.g., numerical dissipation, grid-shock misalignment, and carbuncle phenomena), faced compatibility issues in unstructured CCFVM frameworks due to its reliance on structured grid indices and computationally inefficient and complex triple-matrix Jacobian formulations. To overcome these limitations, three key innovations are introduced: (1) compatibility improvements for unstructured hybrid grids within CCFVM architectures; (2) algebraic reformulation of flux expressions to reduce computational complexity; and (3) an improved shock detector that expands detection zones to mitigate grid-shock misalignment effects. Implemented in the NNW-FlowStar (National numerical wind tunnel) solver, the modified HLLE++ scheme is systematically validated through six test cases spanning subsonic to hypersonic regimes. Numerical results demonstrate four key advantages of the developed methodology: (1) robust compatibility with CCFVM frameworks for unstructured hybrid grid; (2) effective elimination of shock-alignment artifacts through localized dissipation control; (3) excellent applicability for wide-speed-range flow simulations across subsonic-to-hypersonic; and (4) enhanced computational efficiency through optimized flux calculation. This work extends the applicability of HLLE++ to unstructured CCFVM frameworks, offering a novel solution for numerical simulations of complex flow.
Cui et al. (Tue,) studied this question.