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A generalized formulation for gradient schemes in unstructured finite volume method

Mandeep Deka, Ashwani Assam, Ganesh Natarajan

TL;DR

This work introduces a generalized gradient reconstruction framework for unstructured finite volume methods based on a dyadic sum‑vector product. By defining a neighbourhood and two mesh‑dependent geometric vectors per neighbour, it reconstructs centroidal gradients via $\nabla \phi_i = \mathbb{P}^{-1} \sum_f \mathbf{a}_f (\mathbf{b}_f \cdot \nabla \phi_f)$ with $\mathbb{P}=\sum_f (\mathbf{a}_f \otimes \mathbf{b}_f)$, enabling existing GG and LSQ schemes to be viewed as special cases and allowing hybrids and flexible damping. The paper demonstrates how Green–Gauss and least‑squares families arise from particular vector choices, introduces Taylor–Gauss as a hybrid, and then augments the framework with flexible, $\alpha$‑damped variants to improve robustness on distorted meshes. While not prescribing exact vector choices, the framework provides a principled design space for constructing new, potentially more reliable gradient reconstructions in unstructured grids. The approach has potential practical impact for improving stability and accuracy of high‑fidelity CFD solvers on complex geometries, albeit requiring numerical validation for specific applications.

Abstract

We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework by a suitable choice of the geometric vectors that define the dyadic sum tensor. The proposed framework also allows us to re-interpret certain hybrid schemes, which might not be derivable through traditional routes. Additionally, a generalization of flexible gradient schemes is proposed that can be employed to enhance the robustness of consistent gradient schemes without compromising on the accuracy of the computed gradients.

A generalized formulation for gradient schemes in unstructured finite volume method

TL;DR

This work introduces a generalized gradient reconstruction framework for unstructured finite volume methods based on a dyadic sum‑vector product. By defining a neighbourhood and two mesh‑dependent geometric vectors per neighbour, it reconstructs centroidal gradients via with , enabling existing GG and LSQ schemes to be viewed as special cases and allowing hybrids and flexible damping. The paper demonstrates how Green–Gauss and least‑squares families arise from particular vector choices, introduces Taylor–Gauss as a hybrid, and then augments the framework with flexible, ‑damped variants to improve robustness on distorted meshes. While not prescribing exact vector choices, the framework provides a principled design space for constructing new, potentially more reliable gradient reconstructions in unstructured grids. The approach has potential practical impact for improving stability and accuracy of high‑fidelity CFD solvers on complex geometries, albeit requiring numerical validation for specific applications.

Abstract

We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework by a suitable choice of the geometric vectors that define the dyadic sum tensor. The proposed framework also allows us to re-interpret certain hybrid schemes, which might not be derivable through traditional routes. Additionally, a generalization of flexible gradient schemes is proposed that can be employed to enhance the robustness of consistent gradient schemes without compromising on the accuracy of the computed gradients.
Paper Structure (15 sections, 45 equations, 2 figures)

This paper contains 15 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dyadic sum-vector product formalism
  3. Revisiting the existing gradient reconstruction schemes
  4. Green-Gauss schemes
  5. Least squares schemes
  6. Hybrid schemes
  7. Generalization of flexible gradient schemes
  8. A flexible variant of the MGG gradient scheme
  9. A general flexible scheme
  10. General discussions and summary
  11. Error analysis
  12. Term 1:
  13. Term 2:
  14. Term 3:
  15. Some useful identities

Figures (2)

  • Figure 1: Configuration of a cell in an unstructured mesh with a compact "neighbourhood" denoted by a "$\times$" symbol. A representative point in the neighbourhood is denoted by "$f$" where the two geometric vectors are $\mathbf{a}_f$ and $\mathbf{b}_f$ are shown. The points "$f,1$" and "$f,2$" are along the vector $\mathbf{b}_f$.
  • Figure 2: Typical polygonal geometry showing the cells sharing a face. The cells are denoted by their centroids $i$ and $j$, while $f$ represents the face as well as the face center.