Short note: empirical findings for spatial and temporal discretization orders for the Taylor – Green vortex

The purpose of this paper is to shed some light into the equivalency of grid size and spatial and temporal orders of accuracy for turbulent flows.

This paper compared several finite difference schemes on various meshes and various orders of accuracy for the canonical Taylor–Green vortex testcase.

A remarkable empirical result observed is that the accuracy of eighth-order schemes on grids of size 2 h was equivalent to second-order schemes on grids of size h. This was the case when going from 100³ to 200³ as well as from 200³ to 400³. In all these cases, kinetic energy, vorticity and numerical dissipation were compared in detail.

The results obtained were carried out using finite difference solvers. It remains to be seen if similar behaviors are also observed for other numerical schemes.

These (surprising) findings should be of interest to the larger community.