A novel least-squares gradient and interpolation scheme based on the finite volume method

This paper proposes a novel gradient reconstruction scheme, termed the iterative least-squares (ILSQ) reconstruction and an improved gradient interpolation method within the finite volume method (FVM) framework. The improved interpolation scheme is derived from the formulation process of the ILSQ reconstruction. In comparison with the conventional least-squares (LSQ) reconstruction, the ILSQ method enhances accuracy by incorporating gradient information from neighboring cells. This approach effectively emulates a mesh refinement mechanism through a higher-order representation of local solution. The scheme maintains compatibility with compact stencils and is suitable for implicit discretization for pressure-based solvers. Extensive accuracy analyses across various mesh topologies demonstrate that the error of the ILSQ reconstruction is consistently lower than that of the LSQ reconstruction in all test cases, while maintaining good robustness and extensibility. On highly-curved thin grids, where the LSQ reconstruction performs poorly, the ILSQ reconstruction delivers a pronounced improvement. The improved interpolation scheme achieves at least second order accuracy on the non-skewed grids without consuming additional computational resources, and it also outperforms other schemes on skewed grids. The proposed method has been validated through benchmark incompressible and compressible flow cases, demonstrating a certain degree of error reduction and improvement in convergence. Overall, the ILSQ reconstruction and the improved interpolation scheme offer a robust and efficient alternative to existing approaches, combining higher accuracy with practical applicability for CFD problems.