Time-reversibility of the Euler equations as a benchmark for energy conserving schemes

The three-dimensional incompressible Euler equations are time-reversible. This property should be preserved as well as possible by numerical discretizations. This article investigates the time-reversibility properties of various solvers designed for incompressible Navier–Stokes computations. The test case is the inviscid Taylor–Green vortex, which becomes “turbulent” before the time is reversed to try to recover the initial condition. The simulations are performed using high and low order finite difference solvers as well as using a pseudo-spectral solver. Various time-stepping schemes are also investigated. Although the flow statistics are significantly affected by the accuracy of the space discretization, the time-reversibility is not because most space-discretizations are time-reversible for an exact time-stepping. The crucial factor for time-reversibility is the accuracy of the time-stepping scheme and its interaction with the space-discretization. Furthermore, an important practical requirement for the solver is to be energy conserving in order to avoid numerical instability. An energy conserving solver using an accurate time-stepping is then able to go back almost perfectly from a complex “turbulent” flow to the simple initial condition. Therefore, we propose that this constitutes a severe and useful benchmark that Navier–Stokes solvers should challenge. The present investigations and their conclusions are also supported by parallel 1-D investigations, using the non-linear convection equation (inviscid Burgers) and the linear convection equation.