Numerical experiments on the efficiency of local grid refinement based on truncation error estimates

TL;DR

This paper evaluates various local grid refinement criteria based on truncation error estimates in finite volume methods, demonstrating improved efficiency with volume-weighted criteria and higher Reynolds numbers.

Contribution

It introduces and compares multiple truncation error-based refinement criteria, highlighting the effectiveness of volume-weighted approaches in finite volume simulations.

Findings

Volume-weighted truncation error criteria outperform simple error-based criteria.

Refinement efficiency increases with Reynolds number.

Handling of grid interface errors impacts overall refinement performance.

Abstract

Local grid refinement aims to optimise the relationship between accuracy of the results and number of grid nodes. In the context of the finite volume method no single local refinement criterion has been globally established as optimum for the selection of the control volumes to subdivide, since it is not easy to associate the discretisation error with an easily computable quantity in each control volume. Often the grid refinement criterion is based on an estimate of the truncation error in each control volume, because the truncation error is a natural measure of the discrepancy between the algebraic finite-volume equations and the original differential equations. However, it is not a straightforward task to associate the truncation error with the optimum grid density because of the complexity of the relationship between truncation and discretisation errors. In the present work several criteria based on a truncation error estimate are tested and compared on a regularised lid-driven cavity case at various Reynolds numbers. It is shown that criteria where the truncation error is weighted by the volume of the grid cells perform better than using just the truncation error as the criterion. Also it is observed that the efficiency of local refinement increases with the Reynolds number. The truncation error is estimated by restricting the solution to a coarser grid and applying the coarse grid discrete operator. The complication that high truncation error develops at grid level interfaces is also investigated and several treatments are tested.