Unconditional stability of the second-order consistent splitting scheme for the Navier–Stokes equations

Establish whether the second-order consistent splitting scheme introduced in reference [H2004] for the incompressible Navier–Stokes equations is unconditionally stable, thereby resolving the stability limitation associated with its second-order pressure extrapolation.

Background

The paper discusses prior work on consistent splitting schemes for incompressible flow problems. A second-order consistent splitting scheme was previously introduced for the Navier–Stokes equations, but its unconditional stability was not established.

The unresolved issue is linked to the second-order extrapolation of the pressure term, which creates a principal difficulty in proving unconditional stability. The present paper motivates an alternative shifted Taylor-expansion construction for natural convection equations, rather than resolving the cited Navier–Stokes stability question directly.

References

For instance, a second-order consistent splitting scheme was introduced in , which is primarily designed for Navier-Stokes equations; however, whether this second-order scheme is unconditionally stable remains an open problem.

— A new second-order consistent splitting scheme for the Natural Convection equations  (2609.28961 - Si et al., 24 Sep 2026) in Section 1, Introduction