In computational fluid dynamics (CFD), the finite volume method (FVM) has become one of the most widely adopted numerical discretization techniques due to its excellent conservation properties and flexibility in handling complex geometries. Whether in commercial software or open-source solvers, FVM offers an efficient and intuitive framework for solving the Navier–Stokes equations on structured or unstructured meshes. However, the increasing complexity of industrial applications and the growing use of anisotropic or highly skewed grids have raised the demand for improved numerical accuracy, particularly in the computation of spatial derivatives.
Among various discretized terms in the governing equations, gradient reconstruction plays a central role. Accurate estimation of gradients is essential for evaluating convective fluxes, diffusive terms, and pressure gradients—components that are particularly sensitive in pressure-based algorithms. In segregated solvers, gradient errors can lead to pressure–velocity decoupling, while in coupled solvers, gradient approximation directly affects the stability and convergence of the system matrix. Thus, the development of accurate and robust gradient reconstruction schemes has become a critical issue for advancing FVM solvers.
The Green–Gauss (GG) reconstruction is one of the most commonly used gradient estimation approaches within the FVM framework. By applying the Gauss divergence theorem, the GG reconstruction expresses cell-centered gradients using face-normal fluxes. However, because FVM operates primarily on cell-centered values, estimating the field value at face centers requires an interpolation step, the accuracy of which has a profound effect on the overall performance of the GG reconstruction. Traditional schemes—such as inverse-volume weighted interpolation or second-order linear interpolation—offer simplicity and compactness, but they typically suffer from accuracy loss on distorted meshes and often fail to maintain formal second-order convergence in practical cases.
Several improvements to the GG framework have been proposed in recent literature. The Iterative Green–Gauss (IGG) reconstruction introduces a feedback loop where cell gradients are updated iteratively, improving consistency and reducing bias. Meanwhile, Deka et al. [1] proposed the Modified Green–Gauss (MGG) scheme, which replaces traditional interpolation with a more accurate geometric reconstruction, offering better performance on irregular grids. However, both IGG and MGG still rely on interpolation schemes with limited formal accuracy and may incur additional computational cost due to extended stencils or geometric complexity.
To address these challenges, the present study introduces a Modified Interpolation scheme that significantly enhances interpolation accuracy while preserving a compact stencil. The method retains the basic structure of second-order schemes by incorporating gradients from adjacent cells but constructs the interpolation based on enhanced Taylor series expansions in the local direction normal to each face. The resulting expression captures higher-order behavior using only local geometric information, achieving formal fourth-order accuracy. Importantly, it avoids the need for extended stencils or node-based data, making it efficient and easy to implement in existing solvers.
Building upon this interpolation improvement, a new gradient reconstruction method termed Modified Iterative Green–Gauss (MIGG) is proposed. MIGG integrates the Modified Interpolation into the IGG framework, combining the iterative refinement process with a high-accuracy interpolation step. This allows for better face-value approximations during each iteration and results in significantly improved gradient estimates while retaining computational efficiency.
To evaluate the effectiveness of the proposed methods, numerical experiments are conducted on a series of hybrid grids with aspect ratios varying from 1 to 1000 (Fig. 1). These tests include manufactured solutions with known analytical gradients, allowing for quantitative comparison of interpolation and gradient accuracy across different grid resolutions and geometries. The Modified Interpolation scheme alone is shown to achieve fourth-order convergence, vastly outperforming the standard second-order interpolation (Fig. 2 right). The MIGG reconstruction consistently achieves second-order convergence, with substantially lower error magnitudes than both the traditional IGG and the MGG scheme proposed by Deka et al. (Fig. 2 left). These improvements are observed across all grid types, including highly stretched and skewed configurations.
The proposed methods offer several key advantages. First, they maintain a compact stencil, avoiding the complexity of extended-node or multi-level interpolation. Second, they are compatible with both structured and unstructured meshes, making them applicable to a wide range of engineering problems. Third, the high accuracy achieved without additional computational burden makes these schemes especially suitable for large-scale CFD simulations where precision and efficiency are equally important.
In conclusion, this study presents a high-accuracy, computationally efficient interpolation and gradient reconstruction framework for the finite volume method. The Modified Interpolation scheme achieves fourth-order accuracy while retaining simplicity, and its integration into the MIGG reconstruction leads to a robust and accurate gradient reconstruction strategy. These developments provide a valuable contribution to the field of CFD, particularly for simulations involving complex geometries, anisotropic meshes, or strong gradient variations.

Fig. 1 Schematic of the hybrid aspect-ratio grids
 

Finite volume method, Gradient reconstruction, Interpolation