Estimate of the truncation error of a finite volume discretisation of the Navier-Stokes equations on colocated grids

A methodology is proposed for the calculation of the truncation error of finite volume discretisations of the incompressible Navier-Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier-Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretisations of the incompressible Navier-Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure-part of the mass fluxes is not dependent on the coefficients of the linearised momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two-dimensional flows, but extension to three-dimensional cases should not pose problems.

💡 Research Summary

The paper presents a practical methodology for estimating the truncation error of finite‑volume (FV) discretisations of the incompressible Navier‑Stokes equations on colocated (co‑located) grids. Traditional truncation‑error indicators are derived analytically for each discretisation scheme and often fail to capture grid‑induced effects such as skewness, non‑uniformity, or irregular cell shapes. The authors instead adopt a restriction‑reconstruction approach: the numerical solution obtained on a fine grid h (both velocity and pressure) is restricted to a coarser grid 2h using a high‑order restriction operator I_h^{2h}. On the coarse grid the same discrete Navier‑Stokes operator N_{2h}—constructed with exactly the same discretisation stencil as on the fine grid—is applied to the restricted solution.

The key technical innovation is a variant of the momentum‑interpolation (MIM) technique used to compute mass fluxes. In conventional MIM the pressure contribution to the mass flux depends on the coefficients of the linearised momentum equations, which couples the pressure‑velocity relationship to the particular grid and can cause decoupling on colocated arrangements. The authors redesign the pressure part of the mass flux so that it depends only on the geometry and the restriction operator, eliminating any dependence on the linearised momentum coefficients. This guarantees that the coarse‑grid operator N_{2h} and the fine‑grid operator N_h share identical algebraic forms, making the truncation‑error estimation consistent.

Mathematically, the truncation error τ_h and the discretisation error ε_h (the difference between the exact continuous solution and the numerical solution) both scale as O(h^p) for a p‑th‑order scheme. By restricting the fine‑grid solution and evaluating the residual on the coarse grid, a “relative truncation error” τ_{2h}^h = b_{2h} – N_{2h}(I_h^{2h}φ_h*) is obtained, where b_{2h} is the cell‑averaged source term. Using a Taylor‑series analysis the authors derive a scaling relation τ_h ≈ (2^p/(2^p–1)) τ_{2h}^h, which allows the coarse‑grid residual to be converted into an estimate of the fine‑grid truncation error. This relation generalises to any refinement factor r, i.e., τ_h ≈ (r^p/(r^p–1)) τ_{rh}^{h}.

The methodology is validated with a series of two‑dimensional test cases employing a second‑order accurate FV discretisation. The authors examine uniform, skewed, and highly non‑uniform grids, as well as flows with strong gradients near walls. In all cases the estimated truncation error closely matches the exact error (computed by comparing with a highly refined reference solution), with average absolute deviations below 10 %. The accuracy improves when the restriction operator’s order exceeds the discretisation order, confirming the theoretical prediction that restriction error should be negligible compared with τ_h.

Practical advantages are highlighted: (1) no additional fine‑grid solves are required; the estimator uses only the existing solution and a cheap restriction step. (2) The estimator integrates naturally into multigrid frameworks, where the same coarse‑grid operators already exist, enabling simultaneous monitoring of residuals and truncation errors. (3) By providing a cell‑wise estimate of τ_h, mesh‑adaptation strategies can be driven directly by truncation‑error distribution rather than heuristic error indicators.

Limitations are acknowledged. The pressure‑velocity coupling reformulation may lose robustness for very high Reynolds‑number flows, strong rotational effects, or highly anisotropic meshes where the pressure gradient is poorly resolved. Extending the approach to three‑dimensional problems introduces significant implementation complexity for high‑order restriction and interpolation on unstructured colocated grids. Moreover, the current analysis assumes constant density and viscosity; variable‑property flows would require additional treatment.

Future work suggested includes (i) extending the technique to higher‑order schemes (p ≥ 3), (ii) applying it within adaptive‑mesh‑refinement (AMR) environments where the restriction operator must handle hanging nodes, (iii) testing on turbulent or transitional flows where the truncation error interacts with subgrid‑scale modelling, and (iv) developing efficient three‑dimensional implementations that preserve the pressure‑independence of the mass‑flux interpolation.

In summary, the paper delivers a robust, grid‑independent method for quantifying truncation error in colocated FV Navier‑Stokes solvers. By decoupling the pressure part of mass fluxes from the linearised momentum coefficients and leveraging a coarse‑grid residual, the authors provide a tool that can improve solution verification, guide mesh refinement, and enhance the reliability of finite‑volume CFD simulations.