Stability bounds for explicit fractional-step schemes for the Navier–Stokes equations at high Reynolds number

Unwanted spatial oscillations of significant amplitude, but not of the conventional odd–even type, can be generated by an instability inherent in some widely used explicit time-advancement schemes applied to the Navier–Stokes equations. In particular the second-order Gear scheme is susceptible to this instability, as well as the second-order Adams–Bashforth and Runge–Kutta schemes. In its simplest form, the instability introduces slowly growing, divergence-free perturbations to the velocity field. This ‘weakly unstable’ behaviour, whilst known theoretically, can remain concealed in practical simulations and can thus be under-rated. This paper demonstrates the relevance of the instability and highlights the requirement of higher order in an explicit temporal scheme, in order to reduce the affects. In response to this requirement a third-order scheme is developed, that is similar in concept to the second-order Gear scheme. The benefits of this temporal scheme are demonstrated both in reference to numerical stability, and also in relation to dissipation in a co-located spatial scheme.